How to find the equation of a line of which a diameter is on with one given endpoint? Is it possible?

Question

E.g., $A(0,2)$ lies on Circle $C$: $x^2+y^2+3x+4y-12=0$. Line $L$ passes through $A$ and divides the circle into two equal halves. Find the equation of $L$.

Answer 1

Complete the squares to find a form of the equation for the circle $C$

from which it is easy to find the coordinates of its center:

$\left(x+\frac32\right)^2+(y+2)^2=r^2$.

Now you have two points on line $L$.

Can you take it from here?