The mid point of the chord x – 2y + 7 = 0 w.r.t the circle $x^2+y^2-2x-10y+1=0$ is

Equation of circle $x^2+y^2-2x-10y+1=0$

 So; center=(1,5)

 Equation of chord x−2y+7=0

P= mid-point of chord =(h,k)

Now, P must satisfy the equation of chord x−2y+7=0

h−2k+7=0    -  (1)

Now; slope of AB  X slope of PC=−1

 slope of AB=$\frac{1}{2}$(given equation)

  $\Rightarrow \frac{k-5}{h-1}\left(\frac{1}{2}\right)=-1$

$$\begin{gathered} \Rightarrow 2h+k-7=0\to \left(2\right) \\ By solving \left(1\right) and \left(2\right) we get \\ h=\frac{7}{5},k=\frac{21}{5} \end{gathered}$$