{"id":539992,"date":"2023-04-23T01:36:33","date_gmt":"2023-04-22T20:06:33","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/consider-the-family-of-circles-x2y2%e2%88%923x%e2%88%924y%e2%88%92c10-c1%e2%88%88ni123n-also-let-all-circles-intersects-x-axis-at-integral-points-only-and-c1-2\/"},"modified":"2025-06-20T13:27:44","modified_gmt":"2025-06-20T07:57:44","slug":"consider-the-family-of-circles-x2y2%e2%88%923x%e2%88%924y%e2%88%92c10-c1%e2%88%88ni123n-also-let-all-circles-intersects-x-axis-at-integral-points-only-and-c1-2","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/consider-the-family-of-circles-x2y23x4yc10c1ni1\/","title":{"rendered":"Consider the family of circles :\u00a0x2+y2\u22123x\u22124y\u2212c1=0,\u00a0c1\u2208N(i=1,2,3,\u2026n)\u00a0Also, let all circles intersects x-axis at integral points only and\u00a0c1Consider the family of circles : <\/i>x<\/mi>2<\/mn><\/msup>+<\/mo>y<\/mi>2<\/mn><\/msup>\u2212<\/mo>3<\/mn>x<\/mi>\u2212<\/mo>4<\/mn>y<\/mi>\u2212<\/mo>c<\/mi>1<\/mn><\/msub>=<\/mo>0<\/mn><\/math>, c<\/mi>1<\/mn><\/msub>\u2208<\/mo>N<\/mi>(<\/mo>i<\/mi>=<\/mo>1<\/mn>,<\/mo>2<\/mn>,<\/mo>3<\/mn>,<\/mo>\u2026<\/mo>n<\/mi>)<\/mo><\/math> <\/p>

Also, let all circles intersects x-axis at integral points only and <\/i>c<\/mi>1<\/mn><\/msub><<\/mo>c<\/mi>2<\/mn><\/msub><<\/mo>c<\/mi>3<\/mn><\/msub><<\/mo>c<\/mi>4<\/mn><\/msub>\u2026<\/mo><<\/mo>c<\/mi>n<\/mi><\/msub><\/math> . A point (x,y) is said tobe integral point, if both coordinates x and y are integers.<\/i><\/p>","options":null,"solution":"

Step 1: Rewriting the Circle Equation<\/h3>\r\nFirst, we rewrite the circle equation in a more standard form by completing the square:\r\n\r\nx<\/mi>2<\/mn><\/msup>\u2212<\/mo>3<\/mn>x<\/mi>+<\/mo>y<\/mi>2<\/mn><\/msup>\u2212<\/mo>4<\/mn>y<\/mi>=<\/mo>c<\/mi>1<\/mn><\/msub><\/mrow>x^2 - 3x + y^2 - 4y = c_1<\/annotation><\/semantics><\/math><\/span>x<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span>x<\/span>+<\/span><\/span>y<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>4<\/span>y<\/span>=<\/span><\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nCompleting the square for x<\/mi><\/mrow>x<\/annotation><\/semantics><\/math><\/span>x<\/span><\/span><\/span><\/span> and y<\/mi><\/mrow>y<\/annotation><\/semantics><\/math><\/span>y<\/span><\/span><\/span><\/span>:\r\n\r\n(<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>2<\/mn><\/mfrac>)<\/mo>2<\/mn><\/msup>\u2212<\/mo>9<\/mn>4<\/mn><\/mfrac>+<\/mo>(<\/mo>y<\/mi>\u2212<\/mo>2<\/mn>)<\/mo>2<\/mn><\/msup>\u2212<\/mo>4<\/mn>=<\/mo>c<\/mi>1<\/mn><\/msub><\/mrow>(x - \\frac{3}{2})^2 - \\frac{9}{4} + (y - 2)^2 - 4 = c_1<\/annotation><\/semantics><\/math><\/span>(<\/span>x<\/span>\u2212<\/span><\/span>23<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>49<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>(<\/span>y<\/span>\u2212<\/span><\/span>2<\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>4<\/span>=<\/span><\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> (<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>2<\/mn><\/mfrac>)<\/mo>2<\/mn><\/msup>+<\/mo>(<\/mo>y<\/mi>\u2212<\/mo>2<\/mn>)<\/mo>2<\/mn><\/msup>=<\/mo>c<\/mi>1<\/mn><\/msub>+<\/mo>13<\/mn>4<\/mn><\/mfrac><\/mrow>(x - \\frac{3}{2})^2 + (y - 2)^2 = c_1 + \\frac{13}{4}<\/annotation><\/semantics><\/math><\/span>(<\/span>x<\/span>\u2212<\/span><\/span>23<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>(<\/span>y<\/span>\u2212<\/span><\/span>2<\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>413<\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n

Step 2: Conditions for Circle Intersection with the X-Axis<\/h3>\r\nA circle intersects the x-axis where y<\/mi>=<\/mo>0<\/mn><\/mrow>y = 0<\/annotation><\/semantics><\/math><\/span>y<\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>. Substituting y<\/mi>=<\/mo>0<\/mn><\/mrow>y = 0<\/annotation><\/semantics><\/math><\/span>y<\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span> into the modified circle equation gives:\r\n\r\n(<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>2<\/mn><\/mfrac>)<\/mo>2<\/mn><\/msup>+<\/mo>(<\/mo>2<\/mn>)<\/mo>2<\/mn><\/msup>=<\/mo>c<\/mi>1<\/mn><\/msub>+<\/mo>13<\/mn>4<\/mn><\/mfrac><\/mrow>(x - \\frac{3}{2})^2 + (2)^2 = c_1 + \\frac{13}{4}<\/annotation><\/semantics><\/math><\/span>(<\/span>x<\/span>\u2212<\/span><\/span>23<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>(<\/span>2<\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>413<\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> (<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>2<\/mn><\/mfrac>)<\/mo>2<\/mn><\/msup>=<\/mo>c<\/mi>1<\/mn><\/msub>+<\/mo>13<\/mn>4<\/mn><\/mfrac>\u2212<\/mo>4<\/mn><\/mrow>(x - \\frac{3}{2})^2 = c_1 + \\frac{13}{4} - 4<\/annotation><\/semantics><\/math><\/span>(<\/span>x<\/span>\u2212<\/span><\/span>23<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>413<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span> (<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>2<\/mn><\/mfrac>)<\/mo>2<\/mn><\/msup>=<\/mo>c<\/mi>1<\/mn><\/msub>\u2212<\/mo>3<\/mn>4<\/mn><\/mfrac><\/mrow>(x - \\frac{3}{2})^2 = c_1 - \\frac{3}{4}<\/annotation><\/semantics><\/math><\/span>(<\/span>x<\/span>\u2212<\/span><\/span>23<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>43<\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nFor the circle to intersect the x-axis at integral points, the expression (<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>2<\/mn><\/mfrac>)<\/mo>2<\/mn><\/msup><\/mrow>(x - \\frac{3}{2})^2<\/annotation><\/semantics><\/math><\/span>(<\/span>x<\/span>\u2212<\/span><\/span>2<\/span><\/span>3<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> must be such that the solutions for x<\/mi><\/mrow>x<\/annotation><\/semantics><\/math><\/span>x<\/span><\/span><\/span><\/span> are integers. This implies that (<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>2<\/mn><\/mfrac>)<\/mo>2<\/mn><\/msup><\/mrow>(x - \\frac{3}{2})^2<\/annotation><\/semantics><\/math><\/span>(<\/span>x<\/span>\u2212<\/span><\/span>2<\/span><\/span>3<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> itself must be a perfect square.\r\n

Step 3: Ensuring Integral Intersections<\/h3>\r\nThe left side, (<\/mo>x<\/mi>\u2212<\/mo>3<\/mn>2<\/mn><\/mfrac>)<\/mo>2<\/mn><\/msup><\/mrow>(x - \\frac{3}{2})^2<\/annotation><\/semantics><\/math><\/span>(<\/span>x<\/span>\u2212<\/span><\/span>2<\/span><\/span>3<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, can be written as (<\/mo>2<\/mn>x<\/mi>\u2212<\/mo>3<\/mn>)<\/mo>2<\/mn><\/msup>\/<\/mi>4<\/mn><\/mrow>(2x - 3)^2 \/ 4<\/annotation><\/semantics><\/math><\/span>(<\/span>2<\/span>x<\/span>\u2212<\/span><\/span>3<\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/4<\/span><\/span><\/span><\/span>, a transformation of the variable x<\/mi><\/mrow>x<\/annotation><\/semantics><\/math><\/span>x<\/span><\/span><\/span><\/span> into an expression involving integers only when multiplied by 4. So, for integer solutions of x<\/mi><\/mrow>x<\/annotation><\/semantics><\/math><\/span>x<\/span><\/span><\/span><\/span>:\r\n\r\n(<\/mo>2<\/mn>x<\/mi>\u2212<\/mo>3<\/mn>)<\/mo>2<\/mn><\/msup>=<\/mo>4<\/mn>(<\/mo>c<\/mi>1<\/mn><\/msub>\u2212<\/mo>3<\/mn>4<\/mn><\/mfrac>)<\/mo><\/mrow>(2x - 3)^2 = 4(c_1 - \\frac{3}{4})<\/annotation><\/semantics><\/math><\/span>(<\/span>2<\/span>x<\/span>\u2212<\/span><\/span>3<\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>4<\/span>(<\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>43<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> (<\/mo>2<\/mn>x<\/mi>\u2212<\/mo>3<\/mn>)<\/mo>2<\/mn><\/msup>=<\/mo>4<\/mn>c<\/mi>1<\/mn><\/msub>\u2212<\/mo>3<\/mn><\/mrow>(2x - 3)^2 = 4c_1 - 3<\/annotation><\/semantics><\/math><\/span>(<\/span>2<\/span>x<\/span>\u2212<\/span><\/span>3<\/span>)2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>4<\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span>\r\n\r\nThe right side must also be a perfect square for x<\/mi><\/mrow>x<\/annotation><\/semantics><\/math><\/span>x<\/span><\/span><\/span><\/span> to be integral. Let's say 4<\/mn>c<\/mi>1<\/mn><\/msub>\u2212<\/mo>3<\/mn>=<\/mo>k<\/mi>2<\/mn><\/msup><\/mrow>4c_1 - 3 = k^2<\/annotation><\/semantics><\/math><\/span>4<\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span>=<\/span><\/span>k<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> for some integer k<\/mi><\/mrow>k<\/annotation><\/semantics><\/math><\/span>k<\/span><\/span><\/span><\/span>. Solving for c<\/mi>1<\/mn><\/msub><\/mrow>c_1<\/annotation><\/semantics><\/math><\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> gives:\r\n\r\n4<\/mn>c<\/mi>1<\/mn><\/msub>=<\/mo>k<\/mi>2<\/mn><\/msup>+<\/mo>3<\/mn><\/mrow>4c_1 = k^2 + 3<\/annotation><\/semantics><\/math><\/span>4<\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>k<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span> c<\/mi>1<\/mn><\/msub>=<\/mo>k<\/mi>2<\/mn><\/msup>+<\/mo>3<\/mn><\/mrow>4<\/mn><\/mfrac><\/mrow>c_1 = \\frac{k^2 + 3}{4}<\/annotation><\/semantics><\/math><\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>4k<\/span>2<\/span><\/span><\/span><\/span>+<\/span>3<\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nFor c<\/mi>1<\/mn><\/msub><\/mrow>c_1<\/annotation><\/semantics><\/math><\/span>c<\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> to be a natural number, k<\/mi>2<\/mn><\/msup>+<\/mo>3<\/mn><\/mrow>k^2 + 3<\/annotation><\/semantics><\/math><\/span>k<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>3<\/span><\/span><\/span><\/span> must be divisible by 4. Analyzing this condition modulo 4, we observe that k<\/mi>2<\/mn><\/msup><\/mspace><\/mspace>m<\/mi>o<\/mi>d<\/mi><\/mrow>\u2009<\/mtext>\u2009<\/mtext>4<\/mn><\/mrow>k^2 \\mod 4<\/annotation><\/semantics><\/math><\/span>k<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>mod<\/span><\/span>4<\/span><\/span><\/span><\/span> can be either 0 or 1 (since k<\/mi>2<\/mn><\/msup><\/mrow>k^2<\/annotation><\/semantics><\/math><\/span>k<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is the square of either an even or an odd number). The only way k<\/mi>2<\/mn><\/msup>+<\/mo>3<\/mn><\/mrow>k^2 + 3<\/annotation><\/semantics><\/math><\/span>k<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>3<\/span><\/span><\/span><\/span> is divisible by 4 is if k<\/mi>2<\/mn><\/msup><\/mspace><\/mspace>m<\/mi>o<\/mi>d<\/mi><\/mrow>\u2009<\/mtext>\u2009<\/mtext>4<\/mn>=<\/mo>1<\/mn><\/mrow>k^2 \\mod 4 = 1<\/annotation><\/semantics><\/math><\/span>k<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>mod<\/span><\/span>4<\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span>. Therefore, k<\/mi><\/mrow>k<\/annotation><\/semantics><\/math><\/span>k<\/span><\/span><\/span><\/span> must be odd.","subject":"Mathematics"},"yoast_head":"\nConsider the family of circles :\u00a0x2+y2\u22123x\u22124y\u2212c1=0,\u00a0c1\u2208N(i=1,2,3,\u2026n)\u00a0Also, let all circles intersects x-axis at integral points only and\u00a0c1<\/title>\n<meta name=\"description\" content=\"Consider the family of circles :\u00a0x2+y2\u22123x\u22124y\u2212c1=0,\u00a0c1\u2208N(i=1,2,3,\u2026n)\u00a0Also, let all circles intersects x-axis at integral points only and\u00a0c1<c2<c3<c4\u2026<cn\u00a0. A point (x,y) is said tobe integral point, if both coordinates x and y are integers.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/consider-the-family-of-circles-x2y23x4yc10c1ni1\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Consider the family of circles :\u00a0x2+y2\u22123x\u22124y\u2212c1=0,\u00a0c1\u2208N(i=1,2,3,\u2026n)\u00a0Also, let all circles intersects x-axis at integral points only and\u00a0c1\" \/>\n<meta property=\"og:description\" content=\"Consider the family of circles :\u00a0x2+y2\u22123x\u22124y\u2212c1=0,\u00a0c1\u2208N(i=1,2,3,\u2026n)\u00a0Also, let all circles intersects x-axis at integral points only and\u00a0c1<c2<c3<c4\u2026<cn\u00a0. A point (x,y) is said tobe integral point, if both coordinates x and y are integers.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/consider-the-family-of-circles-x2y23x4yc10c1ni1\/\" \/>\n<meta property=\"og:site_name\" content=\"Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/\" \/>\n<meta property=\"article:modified_time\" content=\"2025-06-20T07:57:44+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2025\/04\/infinitylearn.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1920\" \/>\n\t<meta property=\"og:image:height\" content=\"1008\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:site\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Consider the family of circles :\u00a0x2+y2\u22123x\u22124y\u2212c1=0,\u00a0c1\u2208N(i=1,2,3,\u2026n)\u00a0Also, let all circles intersects x-axis at integral points only and\u00a0c1","description":"Consider the family of circles :\u00a0x2+y2\u22123x\u22124y\u2212c1=0,\u00a0c1\u2208N(i=1,2,3,\u2026n)\u00a0Also, let all circles intersects x-axis at integral points only and\u00a0c1<c2<c3<c4\u2026<cn\u00a0. A point (x,y) is said tobe integral point, if both coordinates x and y are integers.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/consider-the-family-of-circles-x2y23x4yc10c1ni1\/","og_locale":"en_US","og_type":"article","og_title":"Consider the family of circles :\u00a0x2+y2\u22123x\u22124y\u2212c1=0,\u00a0c1\u2208N(i=1,2,3,\u2026n)\u00a0Also, let all circles intersects x-axis at integral points only and\u00a0c1","og_description":"Consider the family of circles :\u00a0x2+y2\u22123x\u22124y\u2212c1=0,\u00a0c1\u2208N(i=1,2,3,\u2026n)\u00a0Also, let all circles intersects x-axis at integral points only and\u00a0c1<c2<c3<c4\u2026<cn\u00a0. 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