Question

Let $PQ$ be the common chord of the circles $S_{1}:x^{2}+y^{2}+2x+3y+1=0$ and $S_{2}:x^{2}+y^{2}+4x+3y+2=0$ , then the perimeter (in units) of the triangle $\left(\right.$ where, $\left.C_{1}=\left(-1, \frac{-3}{2}\right)\right)$

• A. $\frac{9}{2}$
• B. $2\sqrt{2}+3$
• C. $3\sqrt{2}+2$
• D. $\frac{3}{2}+2\sqrt{2}$

Answer 1

Answer: (B)

Equation of the common chord is $2x+1=0$

$C_{1}M=\left|\frac{- 2 + 1}{2}\right|=\frac{1}{2}$
$PM=\sqrt{\frac{9}{4} - \frac{1}{4}}=\sqrt{2}$
Length of the common chord $=2\sqrt{2}$
Hence, the perimeter of $\Delta C_{1}PQ=\frac{3}{2}+\frac{3}{2}+2\sqrt{2}$
$=3+2\sqrt{2}$ units