{"id":349680,"date":"2023-01-06T18:59:32","date_gmt":"2023-01-06T13:29:32","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/the-equation-of-the-common-tangent-to-the-parabola-yx2-and-y%e2%88%92x%e2%88%9222-is-are\/"},"modified":"2025-06-26T17:34:35","modified_gmt":"2025-06-26T12:04:35","slug":"the-equation-of-the-common-tangent-to-the-parabola-yx2-and-y%e2%88%92x%e2%88%9222-is-are","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/the-equation-of-the-common-tangent-to-the-parabolayx2and\/","title":{"rendered":"The equation of the common tangent to the parabola\u00a0y=x2\u00a0and\u00a0y=\u2212(x\u22122)2\u00a0is \/are"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"The equation of the common tangent to the parabola\u00a0y=x2\u00a0and\u00a0y=\u2212(x\u22122)2\u00a0is \/are","custom_permalink":"question\/mathematics\/the-equation-of-the-common-tangent-to-the-parabolayx2and\/"},"categories":[],"acf":{"question":"<p>The equation of the common tangent to the parabola&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/math>&nbsp;and&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><msup><mo>)<\/mo><mn>2<\/mn><\/msup><\/math>&nbsp;is \/are<\/p>","options":[{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"right left right left right left right left right left right left\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\"><mtr><mtd><mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/p>","correct":true},{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"right left right left right left right left right left right left\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\"><mtr><mtd><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mtr><\/mtable><\/math><\/p>","correct":false},{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>)<\/mo><\/math><\/p>","correct":false},{"option":"<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>30<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>50<\/mn><\/math><\/p>","correct":false}],"solution":"<p>Any point on the parabola&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>&nbsp;<\/mo><\/math>&nbsp;is&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced separators=\"|\"><mrow><mi>t<\/mi><mo>,<\/mo><mo>&nbsp;<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfenced><\/math>.<\/p><p>Now tangent at&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced separators=\"|\"><mrow><mi>t<\/mi><mo>,<\/mo><mo>&nbsp;<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfenced><\/math>&nbsp;is<\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"left\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\"><mtr><mtd><mi>x<\/mi><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mfenced separators=\"|\"><mrow><mi>y<\/mi><mo>+<\/mo><msub><mi>y<\/mi><mn>1<\/mn><\/msub><\/mrow><\/mfenced><\/mtd><\/mtr><mtr><mtd><mrow><mo stretchy=\"false\">\u21d2<\/mo><mspace width=\"1em\">&nbsp;<\/mspace><\/mrow><mi>t<\/mi><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mfenced separators=\"|\"><mrow><mi>y<\/mi><mo>+<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfenced><\/mtd><\/mtr><mtr><mtd><mrow><mo stretchy=\"false\">\u21d2<\/mo><mspace width=\"1em\">&nbsp;<\/mspace><\/mrow><mn>2<\/mn><mi>t<\/mi><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>\u2212<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mtr><\/mtable><\/math><\/p><p>If it is a tangent to the parabola,&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><msup><mo>)<\/mo><mn>2<\/mn><\/msup><\/math>, then&nbsp;<\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"left\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\"><mtr><mtd><mo>&nbsp;<\/mo><mo>&nbsp;<\/mo><mo>&nbsp;<\/mo><mo>&nbsp;<\/mo><mrow><mn>2<\/mn><mi>t<\/mi><mi>x<\/mi><mo>\u2212<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo>\u2212<\/mo><mo>(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><msup><mo>)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mn>2<\/mn><mi>t<\/mi><mi>x<\/mi><mo>\u2212<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mo>(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>t<\/mi><mo>)<\/mo><mi>x<\/mi><mo>+<\/mo><mfenced separators=\"|\"><mrow><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/mfenced><mo>=<\/mo><mn>0<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/math><\/p><p>Since it has equal roots, so<\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtable columnalign=\"left\" columnspacing=\"1em\"><mtr><mtd><mrow><mo stretchy=\"false\">\u21d2<\/mo><mo>\u2005\u2005\u2005\u2005<\/mo><\/mrow><mi>D<\/mi><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mo>\u2005\u2005\u2005\u2005<\/mo><mrow><mn>4<\/mn><mo>(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>t<\/mi><msup><mo>)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mfenced separators=\"|\"><mrow><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/mfenced><mo>=<\/mo><mn>0<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mrow><mo stretchy=\"false\">\u21d2<\/mo><mo>\u2005\u2005\u2005\u2005<\/mo><\/mrow><mrow><mo>(<\/mo><mn>2<\/mn><mo>\u2212<\/mo><mi>t<\/mi><msup><mo>)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mfenced separators=\"|\"><mrow><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/mfenced><mo>=<\/mo><mn>0<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/math><\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">\u21d2<\/mo><mi>t<\/mi><mo>=<\/mo><mn>2<\/mn><\/math>,&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><\/math><\/p><p>Hence, the equation of the common tangent is &nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/math>,&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/math>&nbsp;.<\/p>","subject":""},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The equation of the common tangent to the parabola\u00a0y=x2\u00a0and\u00a0y=\u2212(x\u22122)2\u00a0is \/are - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"The equation of the common tangent to the parabola\u00a0y=x2\u00a0and\u00a0y=\u2212(x\u22122)2\u00a0is \/are\" \/>\n<meta name=\"robots\" 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