Class 10

Math

All topics

Coordinate Geometry

Find the centre of a circle passing through the points $(6,−6),(3,−7)$ and $(3,3)$.

Let $O(x,y)$ is the center of the circle and $A(6.−6),B(3,−7)$ and $C(3,3)$are the points on the circumference of the circle.

$∴OA=(x_{1}−x_{2})_{2}+(y_{1}−y_{2})_{2} $

$⇒OA=(x−6)_{2}+(y+6)_{2} $

$⇒OB=(x−3)_{2}+(y+7)_{2} $

$⇒OC=(x−3)_{3}+(y−3)_{2} $

$∵$ Radii of the circle are equal

$∴OA=OB$

$⇒(x−6)_{2}+(y+6)_{2} =(x−3)_{2}+(y+7)_{2} $

$⇒x_{2}+36−12x+y_{2}+36+12y=x_{2}+9−6x+y_{2}+49+14y$

$⇒−6x−2y+14=0$

$⇒3x+y=7$....................................................................(i)

Similarly,

$OA=OC$

$⇒(x−6)_{2}+(y+6)_{2} =(x−3)_{2}+(y−3)_{2} $

$⇒x_{2}+36−y+y_{2}+36+12y=x_{2}+9−6x+y_{2}+9−6y$

$⇒−6x+18y+54=0$

$⇒−3x+9y=−27$............................................................(ii)

Adding (i) and (ii)

$⇒10y=−20$

$⇒y=−2$

Substitute the value of $y$ in (i)

$⇒3x+−2=7$

$⇒3x=9$

$⇒x=3$

$∴$ The center of the circle is $(3,−2)$.