Question

Find the locus of mid point of chord of $x^2+y^2+2gx+2fy+c=0$ that pass through the origin.

Answer 1

Let the midpoint of chord be $P(h,k)$

So its eqn: $T=S_1$

$\Rightarrow x(h+g)+y(k+f)-h^2-k^2-gh-fk=0$

{$\because$ eqn of circle $x^2+y^2+2gx+2fy+c=0$]

The circle & chord pass through the part on circle given by:

$x^2+y^2+2gx+2fy+c+l\left[x\right(h+g)+y(k+f)-h^2-k^2-gh-fk]=0$

The chord passes through $(0,0)$

So its center is at $(h,k)$

$\Rightarrow c+l(h^2-k^2-gh-fk)+0+0+0+0=0$

or $-h=g+\lambda \frac{(h+1)}{2}$ & $k=f+\lambda \frac{(k+f)}{2}$

$\therefore l=-2$

$\therefore$ Requried locus of midpoint of chord is:

$2x^2+2y^2+2gh+2fy+c=0$