Question

Two circles centered at $(2, 3)$ and $(5, 6)$ intersect each other. If the radii are equal, the equation of the common chord is ______

• A. x + y - 8 = 0
• B. x - y - 8 = 0
• C. x + y + 1 = 0
• D. x - y + 1 = 0

Answer 1

Answer: (A)

x + y - 8 = 0

Answer 2

Let the radius of both circles are ' $r$ '.
Now, equation of circle with centre at $(2,3)$ is
$S_{1} \equiv(x-2)^{2}+(y-3)^{2}=r^{2}$
and equation of circle with centre at $(5,6)$ is
$S_{2} \equiv(x-5)^{2}+(y-6)^{2}=r^{2}$
Now, the equation common chord
$\equiv$ Radical axis of $S_{1}$ and $S_{2}=0$
$\equiv\left(S_{1}-S_{2}\right)=0$
$\equiv\left[(x-2)^{2}\right]+\left[(y-3)^{2}\right]$
$\equiv x^{2}+y^{2}+4-4 x+9-6 x$
$-x^{2}-y^{2}-25-36+10 x+12 y=0$
$\equiv 6 x+6 y-48=0$
Common chord $\equiv x+y-8=0$