Question

The coordinates of the middle point of the chord which the circle $x^2+y^2+4x-2y-3=0$ cuts off on the line $y=x+2$, are

• A. $\left(\frac{-3}{2},\frac{1}{2}\right)$
• B. $\left(\frac{3}{2},\frac{1}{2}\right)$
• C. $\left(\frac{-3}{2},\frac{-1}{2}\right)$
• D. $\left(\frac{3}{2},\frac{-1}{2}\right)$

Answer 1

Answer: (A)

Equation of chord PQ is $y=x+2$
or $x-y+2=0...\left(1\right)$

Centre of circle is C $\left(-2,1\right)$.
Draw CM $\perp$ PQ, then M is the mid point of PQ.
Equation of any line $\perp$ to PQ is x + y + k = 0
If it passes through C $\left(-2,1\right)$ then $-2+1+k=0$ or $k=1$
Equation of CM is $x+y+1=0...\left(2\right)$
Solving $\left(1\right)$ and $\left(2\right)$, we obtain $x=-\frac{3}{2}$ and $y=\frac{1}{2}$.
$\therefore$ Coordinates of M are $\left(\frac{-3}{2},\frac{1}{2}\right)$