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)

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 $S_2=0$
$\equiv \left(S_1-S_2\right)=0$
$\equiv \left[(x-2{)}^2\right]+\left[(y-3{)}^2\right]$
$\equiv x^2+y^2+4-4x+9-6x$
$-x^2-y^2-25-36+10x+12y=0$
$\equiv 6x+6y-48=0$
Common chord $\equiv x+y-8=0$