{"id":9888,"date":"2022-01-31T21:54:57","date_gmt":"2022-01-31T16:24:57","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9888"},"modified":"2022-01-31T21:56:03","modified_gmt":"2022-01-31T16:26:03","slug":"pole-and-polar-of-a-circle-equation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/pole-and-polar-of-a-circle-equation\/","title":{"rendered":"Pole and Polar of a Circle Equation"},"content":{"rendered":"

Here you will learn what is the pole and polar of a circle and pole of given line with respect to a circle.<\/p>\n

Let’s begin –<\/p>\n

Pole and Polar of a Circle<\/h2>\n

Let any straight line through the given point A\\((x_1, y_1)\\) intersects the given circle S = 0 in two points P and Q and if the tangent of the circle at P and Q meet at the point R then the locus of point R is called polar of point A<\/strong> and point A is called the pole<\/strong>, with respect to the given circle.<\/p>\n

Equation of the Polar of Point<\/h2>\n

The equation of the polar of point \\((x_1, y_1)\\) with respect to circle \\(x^2 + y^2\\) = \\(a^2\\) is\"pole<\/p>\n

\\(xx_1 + yy_1\\) = \\(a^2\\)<\/p><\/blockquote>\n

Proof<\/strong> : Let APQ is a chord given in figure which passes through the point A\\((x_1, y_1)\\) which intersects the circle at points P and Q and the tangents are drawn at points P and Q meet at point R (h, k) then the equation of PQ the chord of contact<\/a> is \\(x_1h + y_1k\\) = \\(a^2\\)<\/p>\n

\\(\\therefore\\)\u00a0 Locus of point S(h, k) is \\(xx_1 + yy_1\\) = \\(a^2\\)\u00a0 which is the equation of the polar.<\/p>\n

Note<\/strong> :<\/p>\n

(i)\u00a0 If point is outside the circle then the equation of polar and chord of contact<\/a> is same. So the chord of contact is polar.<\/p>\n

(ii)\u00a0 If point is inside the circle then chord of contact<\/a> does not exist but polar exist.<\/p>\n

(iii)\u00a0 If point lies on the circle then polar, chord of contact<\/a> and tangent on that point are same.<\/p>\n

(iv)\u00a0 If polar of P w.r.t a circle passes through the point Q, then the polar of point Q will pass through P and hence P and Q are conjugate points of each other w.r.t the given circle.<\/p>\n

(v)\u00a0 If pole of a line w.r.t a circle lies on second line. Then pole of second line lies on first line and hence both lines are conjugate lines of each other w.r.t the given circle.<\/p>\n

Pole of a given line with respect to a circle<\/h2>\n

To find the pole of a line we assume the coordinates of the pole then from these coordinates we find the polar. This polar and the given line represent the same line.<\/p>\n

Then by comparing the coefficients of similar terms we can get the coordinates of the pole.<\/p>\n

The pole of line lx + my + n = 0 w.r.t circle \\(x^2 + y^2\\) = \\(a^2\\)\u00a0 will be<\/p>\n

\\(({-la^2\\over n}, {-ma^2\\over n})\\)<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the pole and polar of a circle and pole of given line with respect to a circle. Let’s begin – Pole and Polar of a Circle Let any straight line through the given point A\\((x_1, y_1)\\) intersects the given circle S = 0 in two points P and Q …<\/p>\n

Pole and Polar of a Circle Equation<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[23],"tags":[],"yoast_head":"\nPole and Polar of a Circle Equation - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the pole and polar of a circle and pole of given line with respect to a circle.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/pole-and-polar-of-a-circle-equation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Pole and Polar of a Circle Equation - 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