Question

Let the mirror image of a circle c1 :x2 + y2 – 2x – 6y + α = 0 in line y = x + 1 be c2 : 5x2 + 5y2 + 10gx + 10fy + 38 = 0. If r is the radius of circle c2, then α + 6r2 is equal to _________.

Answer 1

The correct answer is 12
c1: x2 + y2 – 2x – 6y + α = 0
Then centre = (1, 3) and
radius $(r)=\sqrt{10−α}$
Image of (1, 3) w.r.t. line x – y + 1 = 0 is (2, 2)
c2: 5x2 + 5y2 + 10gx + 10fy + 38 = 0
or
$x^2+y^2+2gx+2fy+\frac{38}{5}=0$
Then (–g, –f) = (2, 2)
∴ g = f = – 2 …(i)
Radius of $c_2=r=\sqrt{4+4−\frac{38}{5}}=\sqrt{10−α}$
$⇒ \frac{2}{5}=10−α$
$∴ α=\frac{48}{5}$ and $r=\sqrt{\frac{2}{5}}$
$∴ α+6r^2=\frac{48}{5}+\frac{12}{5}$
= 12