Question

The equation of the locus of the mid-points of the chords of the circle $4x^2+4y^2-12x+4y+1=0$ that subtend an angle of $2\pi /3$ at its centre is:

Answer 1

The centre of the given circle is $(3/2,-1/2)$ and its radius is $3/2$. From the figure if $M(h,k)$ be the middle point of chord $AB$ subtending an angle $2\pi /3$ at $C$, then
$\frac{CM}{AC}=\cos \frac{\pi }{3}=\frac{1}{2}$ or $4C{M}^2=A{C}^2$
or $4\left[\right(h-3/2{)}^2+(k+1/2{)}^2]=9/4$
$\therefore$ Locus is $4x^2+4y^2-12x+4y+\left(31/4\right)=0$.