The locus of the middle points of chords of the 

circle $x^2+y^2=25$ which are  parallel to the line    $x-2y+3=0$ is

Let $(h,k)$  be the mid-point of a chord of the circle

$x^2+y^2=25$ Then, equation of the chord is

$hx+ky-25=h^2+k^2-25\Rightarrow hx+ky=h^2+k^2$

If this is parallel to $x-2y+3=0,$then 

$-\frac{h}{k}=\frac{1}{2}\Rightarrow 2h+k=0$

So, the locus of  $(h,k)$ is $2x+y=0$