Question

Centre of the circle passing through the points (6,-6), (3,-7) and (3,3) is .

• A. (3 , 2)
• B. (2 , 3)
• C. (3 , -2)

Answer 1

The correct option is C (3 , -2)

Let the centre of the circle be P(x,y).
Distance from P to A,B and C is given by
AP =$\sqrt{(x-6{)}^2+(y+6{)}^2}$
BP = $\sqrt{(x-3{)}^2+(y+7{)}^2}$
CP=$\sqrt{(x-3{)}^2+(y-3{)}^2}$

PA = PB (Radii of the same circle)
$\sqrt{(x-6{)}^2+(y+6{)}^2}=\sqrt{(x-3{)}^2+(y+7{)}^2}$
$\Rightarrow x^2+36-12x+y^2+36+12y=x^2+9-6x+y^2+49+14y$
$\Rightarrow -6x-2y+14=0\to 3x+y=7-----\left(i\right)$
Similarly, PA = PC
$\sqrt{(x-6{)}^2+(y+6{)}^2}=\sqrt{(x-3{)}^2+(y-3{)}^2}$
$\Rightarrow x^2+36-12x+y^2+36+12y=x^2+9-6x+y^2+9-6y$
$\Rightarrow -6x+18y+54=0\to -3x+9y=-27-----\left(ii\right)$

On adding (i) and (ii),
we get 10y = -20 $\Rightarrow y=-2$ and $x=3$
$\therefore$ The centre of the circle is (3 , -2).