A circle passes through (-2, 4) and touches the$y-axis$ at (0, 2). Which one of the following equations can represent a diameter of this circle?

Equation of circle with centre $(h,k)$ and touches

$y-axis$ is given by $x^2+y^2-2hx-2ky+k^2=0$

Since, it touches $y\text{-axis at }(0,2)\therefore k=2_{ }$

$\Rightarrow x^2+y^2-2hx-4y+4=0$

Also, it passes through (-2, 4) 

$\begin{array}{l}(-2{)}^2+4^2-2h(-2)-4(4)+4=0\\ \Rightarrow 4+16+4h-16+4=0\Rightarrow h=-2\end{array}$

Hence, centre of circle is (-2, 2) 

$(-2,2)$ satisfy the equation given in option (a).

So, diameter of circle is $2x-3y+10=0$.