Question

Find the equation of circle which passes through points (5,-8),(2,-9) and(2,1). Find also the coordinates of its centre and radius

Answer 1

$\textsf{Let the equation of the required circle be x^2+y^2+2gx+2fy+c=0 }$

$\textsf{since the given circle passes through (5,-8), (2,-9) and (2,1), we have}$

$\mathsf{5^2+(-8)^2+2g(5)+2f(-8)+c=0}$

$\implies\mathsf{10g-16f+c=-89}$........(1)

$\mathsf{2^2+(-9)^2+2g(2)+2f(-9)+c=0}$

$\implies\mathsf{4g-18f+c=-85}$........(2)

$\mathsf{2^2+(1)^2+2g(2)+2f(1)+c=0}$

$\implies\mathsf{4g+2f+c=-5}$........(3)

$\mathsf{(1)-(2), gives}$

$\mathsf{10g-16f+c=-89}$......(1)

$\mathsf{4g-18f+c=-85}$........(2)

$\mathsf{6g+2f=-4}$

$\mathsf{3g+f=-2}$........(4)

$\mathsf{(2)-(3), gives}$

$\mathsf{4g-18f+c=-85}$........(2)

$\mathsf{4g+2f+c=-5}$........(3)

$\mathsf{-20f=-80}$

$\mathsf{g-f=-9}$

$\implies\mathsf{f=4}$

$\textsf{put f=4 in (4), we get}$

$\mathsf{3g+4=-2}$

$\mathsf{3g=-6}$

$\implies\mathsf{g=-2}$

$\textsf{put g=-2and f=4 in (3), we get}$

$\mathsf{4(-2)+2(4)+c=-5}$

$\implies\mathsf{c=-5}$

$\textsf{The equation of the required circle is}$

$\mathsf{x^2+y^2+2(-2)x+2(4)y-5=0}$

$\implies\boxed{\mathsf{x^2+y^2-4x+8y-5=0}}$

$\textsf{Its centre is (-g,-f)=(2,-4)}$

$\textsf{Radius}\mathsf{=\sqrt{g^2+f^2-c}}$

$\textsf{Radius,}\mathsf{=\sqrt{4+16+5}}$

$\textsf{Radius,}\mathsf{=\sqrt{25}}$

$\implies\textsf{Radius=5 units}$

Find more:

Find the equation of the circle passing through the points (1,2),(3,-4)and (5,-6)find the centre and rafius in each case

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