{"id":539994,"date":"2023-04-23T01:36:49","date_gmt":"2023-04-22T20:06:49","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/if-the-tangent-drawn-at-any-point-p-acos4%e2%81%a1%ce%b8asin4%e2%81%a1%ce%b8-on-the-curve-xya-meets-the-co-ordinate-axes-in-a-b-respectively-then-2\/"},"modified":"2025-06-20T13:27:46","modified_gmt":"2025-06-20T07:57:46","slug":"if-the-tangent-drawn-at-any-point-p-acos4%e2%81%a1%ce%b8asin4%e2%81%a1%ce%b8-on-the-curve-xya-meets-the-co-ordinate-axes-in-a-b-respectively-then-2","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/if-the-tangent-drawn-at-any-point-p-acos4asin4-on\/","title":{"rendered":"If the tangent drawn at any point P \u00a0\u00a0a(cos4\u2061\u03b8,asin4\u2061\u03b8)\u00a0 \u00a0on the curve \u00a0x+y=a\u00a0 meets the co-ordinate axes in A, B respectively then."},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"If the tangent drawn at any point P \u00a0\u00a0a(cos4\u2061\u03b8,asin4\u2061\u03b8)\u00a0 \u00a0on the curve \u00a0x+y=a\u00a0 meets the co-ordinate axes in A, B respectively then.","custom_permalink":"question\/mathematics\/if-the-tangent-drawn-at-any-point-p-acos4asin4-on\/"},"categories":[],"acf":{"question":"<p>If the tangent drawn at any point P &nbsp;&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>a<\/mi><mrow><mo>(<\/mo><msup><mi>cos<\/mi><mn>4<\/mn><\/msup><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo>,<\/mo><mi>a<\/mi><msup><mi>sin<\/mi><mn>4<\/mn><\/msup><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo>)<\/mo><\/mrow><\/math>&nbsp; &nbsp;on the curve &nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msqrt><mi>x<\/mi><\/msqrt><mo>+<\/mo><msqrt><mi>y<\/mi><\/msqrt><mo>=<\/mo><msqrt><mi>a<\/mi><\/msqrt><\/math>&nbsp; meets the co-ordinate axes in A, B respectively then.<\/p>","options":null,"solution":"<div class=\"min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&amp;]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"99a229cd-9def-4fe6-acd3-03efbda820fd\">\r\n<div class=\"flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]\">\r\n<div class=\"markdown prose w-full break-words dark:prose-invert dark\">\r\n<ol>\r\n \t<li><strong>The Curve:<\/strong> The given curve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mi>x<\/mi><\/msqrt><mo>+<\/mo><msqrt><mi>y<\/mi><\/msqrt><mo>=<\/mo><msqrt><mi>a<\/mi><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{x} + \\sqrt{y} = \\sqrt{a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span> is a standard form where <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> is a constant. This represents a quarter-circle in the first quadrant of the Cartesian plane, with the radius <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mi>a<\/mi><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/li>\r\n \t<li><strong>Point P on the Curve:<\/strong> The point <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo separator=\"true\">,<\/mo><mi>a<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">P(a \\cos 4\\theta, a \\sin 4\\theta)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">cos<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">sin<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> is assumed to lie on this curve. However, for it to lie on the curve, the relationship between <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span> coordinates given by <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mi>x<\/mi><\/msqrt><mo>+<\/mo><msqrt><mi>y<\/mi><\/msqrt><mo>=<\/mo><msqrt><mi>a<\/mi><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{x} + \\sqrt{y} = \\sqrt{a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span> should be satisfied. Let\u2019s verify if <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> does lie on the curve or not.<\/li>\r\n \t<li><strong>Tangent at Point P:<\/strong> If <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> indeed lies on the curve, we will need the slope of the tangent line at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> to find where this line intersects the coordinate axes.<\/li>\r\n<\/ol>\r\nLet's start by verifying if <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo separator=\"true\">,<\/mo><mi>a<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">P(a \\cos 4\\theta, a \\sin 4\\theta)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">cos<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">sin<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> lies on the curve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mi>x<\/mi><\/msqrt><mo>+<\/mo><msqrt><mi>y<\/mi><\/msqrt><mo>=<\/mo><msqrt><mi>a<\/mi><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{x} + \\sqrt{y} = \\sqrt{a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&amp;]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"8d7e0147-6083-4873-91dd-0a044cba0592\">\r\n<div class=\"flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]\">\r\n<div class=\"markdown prose w-full break-words dark:prose-invert dark\">\r\n\r\nUpon simplifying the expression, we find <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mi>a<\/mi><\/msqrt><mo stretchy=\"false\">(<\/mo><msqrt><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/msqrt><mo>+<\/mo><msqrt><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/msqrt><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{a}(\\sqrt{\\sin(4\\theta)^2} + \\sqrt{\\cos(4\\theta)^2} - 1)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mop\">sin<\/span><span class=\"mopen\">(<\/span>4<span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mclose\">)<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mop\">cos<\/span><span class=\"mopen\">(<\/span>4<span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mclose\">)<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>. Since <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b8<\/mi><mo>+<\/mo><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>\u03b8<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^2\\theta + \\cos^2\\theta = 1<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop\">sin<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mop\">cos<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> for any <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\theta<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">\u03b8<\/span><\/span><\/span><\/span>, the expression simplifies further to 0, indicating that point <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo separator=\"true\">,<\/mo><mi>a<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">P(a \\cos 4\\theta, a \\sin 4\\theta)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">cos<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">sin<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> does indeed lie on the curve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mi>x<\/mi><\/msqrt><mo>+<\/mo><msqrt><mi>y<\/mi><\/msqrt><mo>=<\/mo><msqrt><mi>a<\/mi><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{x} + \\sqrt{y} = \\sqrt{a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>.\r\n\r\nNext, we need to find the slope of the tangent at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> on the curve. This requires us to differentiate the curve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msqrt><mi>x<\/mi><\/msqrt><mo>+<\/mo><msqrt><mi>y<\/mi><\/msqrt><mo>=<\/mo><msqrt><mi>a<\/mi><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\sqrt{x} + \\sqrt{y} = \\sqrt{a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span> implicitly to find <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{dy}{dx}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">d<\/span><span class=\"mord mathnormal mtight\">x<\/span><\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">d<\/span><span class=\"mord mathnormal mtight\">y<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, and then use the chain rule to evaluate it at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span>. Let's proceed with this. \u200b<button><\/button>\u200b\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&amp;]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"576b2e68-b7d9-426d-810f-6562ea3f58d1\">\r\n<div class=\"flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]\">\r\n<div class=\"markdown prose w-full break-words dark:prose-invert dark\">\r\n\r\nThe expression for the slope of the tangent at point <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> simplifies to <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u2212<\/mo><mfrac><msqrt><mi>x<\/mi><\/msqrt><mrow><msqrt><mi>a<\/mi><\/msqrt><msqrt><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">-\\frac{\\sqrt{x}}{\\sqrt{a} \\sqrt{\\sin(4\\theta)^2}}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord sqrt mtight\"><span class=\"svg-align\"><span class=\"mord mathnormal mtight\">a<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"mord sqrt mtight\"><span class=\"svg-align\"><span class=\"mop mtight\"><span class=\"mtight\">s<\/span><span class=\"mtight\">i<\/span><span class=\"mtight\">n<\/span><\/span><span class=\"mopen mtight\">(<\/span>4<span class=\"mord mathnormal mtight\">\u03b8<\/span><span class=\"mclose mtight\">)<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"sizing reset-size3 size1 mtight\">2<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord sqrt mtight\"><span class=\"svg-align\"><span class=\"mord mathnormal mtight\">x<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, but it should be evaluated further using the x-coordinate of <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span>, which is <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a \\cos 4\\theta<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">cos<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><\/span><\/span><\/span>. Let's correct and evaluate this expression properly using the correct substitution. \u200b<button><\/button>\u200b\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"min-h-[20px] text-message flex flex-col items-start whitespace-pre-wrap break-words [.text-message+&amp;]:mt-5 juice:w-full juice:items-end overflow-x-auto gap-2\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"443e114c-465d-4329-94aa-6abe09da5c74\">\r\n<div class=\"flex w-full flex-col gap-1 juice:empty:hidden juice:first:pt-[3px]\">\r\n<div class=\"markdown prose w-full break-words dark:prose-invert dark\">\r\n\r\nThe slope of the tangent line at point <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span> on the curve simplifies to <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u2212<\/mo><mfrac><msqrt><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/msqrt><msqrt><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/msqrt><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">-\\frac{\\sqrt{\\cos(4\\theta)^2}}{\\sqrt{\\sin(4\\theta)^2}}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord sqrt mtight\"><span class=\"svg-align\"><span class=\"mop mtight\"><span class=\"mtight\">s<\/span><span class=\"mtight\">i<\/span><span class=\"mtight\">n<\/span><\/span><span class=\"mopen mtight\">(<\/span>4<span class=\"mord mathnormal mtight\">\u03b8<\/span><span class=\"mclose mtight\">)<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"sizing reset-size3 size1 mtight\">2<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord sqrt mtight\"><span class=\"svg-align\"><span class=\"mop mtight\"><span class=\"mtight\">c<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\">s<\/span><\/span><span class=\"mopen mtight\">(<\/span>4<span class=\"mord mathnormal mtight\">\u03b8<\/span><span class=\"mclose mtight\">)<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"sizing reset-size3 size1 mtight\">2<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, which further simplifies to <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u2212<\/mo><mfrac><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><\/mrow><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">-\\frac{\\cos 4\\theta}{\\sin 4\\theta}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mtight\">s<\/span><span class=\"mtight\">i<\/span><span class=\"mtight\">n<\/span><\/span>4<span class=\"mord mathnormal mtight\">\u03b8<\/span><\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mtight\">c<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\">s<\/span><\/span>4<span class=\"mord mathnormal mtight\">\u03b8<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> or <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u2212<\/mo><mi>cot<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-\\cot 4\\theta<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mop\">cot<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><\/span><\/span><\/span>. This is the slope of the tangent line at <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><\/span><\/span><\/span>.\r\n\r\nWith the slope <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>cot<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m = -\\cot 4\\theta<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">m<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mop\">cot<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><\/span><\/span><\/span> and the point <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo separator=\"true\">,<\/mo><mi>a<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">P(a \\cos 4\\theta, a \\sin 4\\theta)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">cos<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">sin<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>, the equation of the tangent line can be written in point-slope form: <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>sin<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>cot<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mi>cos<\/mi><mo>\u2061<\/mo><mn>4<\/mn><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">y - a \\sin 4\\theta = -\\cot 4\\theta (x - a \\cos 4\\theta)<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">sin<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mop\">cot<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mop\">cos<\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","subject":"Mathematics"},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - 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