Question

Through a fixed point $(x_1,y_1)$ secants are drawn to the circle $x^2+y^2=a^2$. Show that locus of mid-points of the secants intercepts by the given circle is $x^2+y^2=x{x}_1+y{y}_1$.

Answer 1

If the chord passes through a fixed point $(x_1,y_1)$, then $h{x}_1+k{y}_1=h^2+k^2$.
Hence the locus of the mid-point in this case is
$x{x}_1+y{y}_1=x^2+y^2$ or $x^2+y^2-x{x}_1-y{y}_1=0$
Above represents a circle whose centre is $(\frac{1}{2}x,\frac{1}{2}y)$.