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Q.
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
Conic Sections
Solution:
Equation of chord PQ is $y = x + 2$
or $x - y + 2 = 0 \quad...\left(1\right)$
Centre of circle is C $\left(-2, 1\right)$.
Draw CM $\bot$ PQ, then M is the mid point of PQ.
Equation of any line ${\bot}$ 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 $x + y + 1 = 0 \quad...\left(2\right)$
Solving $\left(1\right)$ and $\left(2\right)$, we obtain $x = -\frac{3}{2}$ and $y = \frac{1}{2}$.
$\therefore $ Coordinates of M are $\left(\frac{-3}{2}, \frac{1}{2}\right)$
