Question

The circle through the points (2,3), (2,2), (3,2)

is
????? ​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{Points are (2,3), (2,2) and (3,2)}$

$\underline{\textbf{To find:}}$

$\textsf{The equation of the circle passing through the points}$

$\mathsf{(2,3),(2,2)\;and\;(3,2)}$

$\underline{\textbf{Solution:}}$

$\textsf{We can find a circle passing through any three non-collinear points}$

$\textsf{Let the equation of the requiredcircle be}\;\mathsf{x^2+y^2+2g\,x+2f\,y+c=0}$-------(1)

$\textsf{Since the circle (1) passes through (2,3), we have}$

$\mathsf{2^2+3^2+2g(2)+2f(3)+c=0}$

$\mathsf{4+9+4g+6f+c=0}$

$\mathsf{4g+6f+c=-13}$--------(2)

$\textsf{Since the circle (1) passes through (2,2), we have}$

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

$\mathsf{4+4+4g+4f+c=0}$

$\mathsf{4g+4f+c=-8}$--------(3)

$\textsf{Since the circle (1) passes through (3,2), we have}$

$\mathsf{3^2+2^2+2g(3)+2f(2)+c=0}$

$\mathsf{9+4+6g+4f+c=0}$

$\mathsf{6g+4f+c=-13}$--------(4)

$\underline{\textsf{Solving (2) and (3):}}$

$\mathsf{(2)-(3)\;\implies}$

$\mathsf{4g+6f+c=-13}$--------(2)

$\mathsf{4g+4f+c=-8}$--------(3)

$\implies\mathsf{2f=-5}$

$\implies\boxed{\mathsf{f=\dfrac{-5}{2}}}$

$\underline{\textsf{Solving (3) and (4):}}$

$\mathsf{(3)-(4)\;\implies}$

$\mathsf{4g+4f+c=-8}$--------(3)

$\mathsf{6g+4f+c=-13}$--------(4)

$\implies\mathsf{-2g=5}$

$\implies\boxed{\mathsf{g=\dfrac{-5}{2}}}$

$\mathsf{Using\;f=\dfrac{-5}{2}\;and\;g=\dfrac{-5}{2}\;in\;(3)}$

$\mathsf{4\left(\dfrac{-5}{2}\right)+4\left(\dfrac{-5}{2}\right)+c=-8}$

$\mathsf{2(-5)+2(-5)+c=-8}$

$\mathsf{-20+c=-8}$

$\mathsf{c=20-8}$

$\implies\boxed{\mathsf{c=12}}$

$\textsf{Equation of the required circle is}$

$\mathsf{x^2+y^2+2\left(\dfrac{-5}{2}\right)\,x+2\left(\dfrac{-5}{2}\right)\,y+12=0}$

$\boxed{\bf\,x^2+y^2-5x-5y+12=0}$

$\underline{\textbf{Find more:}}$

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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Find the equation of circle passing through the points (1,-4) ,(5,2) and having its centre on the line x-2y+9=0

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