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

Answer 1

The correct option is A $(-\frac{3}{2},\frac{1}{2})$

Equation of chord $PQ$ is $y=x+2\Rightarrow y=x+2$ ...(1)

Center of circle is $C(-2,1)$

Draw $CM\perp PQ$, then $M$ is the mid pointg of $PQ$.

Equation of any line $\perp$ to $PQ$ is $x+y+k=0$

If it passes though $C(-2,1)$ then

$-2+1+k=0$ or $k=1$

$\therefore$ Equation of $CM$ is $x+y+1=0$ ...(2)

Solving (1) and (2), we obtain $x=-\frac{3}{2}$ and $y=\frac{1}{2}$

$\therefore$ Coordinates of $M$ are $(-\frac{3}{2},\frac{1}{2})$