How do we find equation of tangent and normal to a circle at a given point?

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How do we find equation of tangent and normal to a circle at a given point?

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Answer 1 · 2017-04-11T17:05:18

Equation of tangent at $(x_{1}, y_{1})$ $(x_1,y_1)$ at circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ $x^2+y^2+2gx+2fy+c=0$ is $x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$ $x x_1+yy_1+g(x+x_1)+f(y+y_1)+c=0$ and equation of normal is $(y_{1} + f) x - (x_{1} + g) y - f x_{1} + g y_{1} = 0$ $(y_1+f)x-(x_1+g)y-fx_1+gy_1=0$

Explanation:

Let the circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ $x^2+y^2+2gx+2fy+c=0$ ,

let us seek normal and tangent at $(x_{1}, y_{1})$ $(x_1,y_1)$

(note that $x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c = 0$ $x_1^2+y_1^2+2gx_1+2fy_1+c=0$ ) ...(A)

as the center of the circle is $(- g, - f)$ $(-g,-f)$ ,

as normal joins $(- g, - f)$ $(-g,-f)$ and $(x_{1}, y_{1})$ $(x_1,y_1)$ its slope is $\frac{y_{1} + f}{x_{1} + g}$ $(y_1+f)/(x_1+g)$

and equation of normal is $y - y_{1} = \frac{y_{1} + f}{x_{1} + g} (x - x_{1})$ $y-y_1=(y_1+f)/(x_1+g)(x-x_1)$

i.e. $(x_{1} + g) y - x_{1} y_{1} - g y_{1} = (y_{1} + f) x - x_{1} y_{1} - f x_{1}$ $(x_1+g)y-x_1y_1-gy_1=(y_1+f)x-x_1y_1-fx_1$

or $(y_{1} + f) x - (x_{1} + g) y - f x_{1} + g y_{1} = 0$ $(y_1+f)x-(x_1+g)y-fx_1+gy_1=0$

and as product of slopes of normal and tangent is $- 1$ $-1$ ,

slope of tangent is $- \frac{x_{1} + g}{y_{1} + f}$ $-(x_1+g)/(y_1+f)$ and its equation will be

$y - y_{1} = - \frac{x_{1} + g}{y_{1} + f} (x - x_{1})$ $y-y_1=-(x_1+g)/(y_1+f)(x-x_1)$

or $(y - y_{1}) (y_{1} + f) + (x - x_{1}) (x_{1} + g) = 0$ $(y-y_1)(y_1+f)+(x-x_1)(x_1+g)=0$

or $y y_{1} - y_{1}^{2} + f y - f y_{1} + x x_{1} - x_{1}^{2} + g x - g x_{1} = 0$ $yy_1-y_1^2+fy-fy_1+x x_1-x_1^2+gx-gx_1=0$

or $x x_{1} + y y_{1} + g x + f y - (x_{1}^{2} + y_{1}^{2} + f y_{1} + g x_{1}) = 0$ $x x_1+yy_1+gx+fy-(x_1^2+y_1^2+fy_1+gx_1)=0$

from (A) $x_{1}^{2} + y_{1}^{2} + f y_{1} + g x_{1} = - f y_{1} - g x_{1} - c$ $x_1^2+y_1^2+fy_1+gx_1=-fy_1-gx_1-c$

Hence equation of tangent is $x x_{1} + y y_{1} + g x + f y + f y_{1} + g x_{1} + c = 0$ $x x_1+yy_1+gx+fy+fy_1+gx_1+c=0$

or $x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$ $x x_1+yy_1+g(x+x_1)+f(y+y_1)+c=0$

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How do we find equation of tangent and normal to a circle at a given point?

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