Find the coordinates of a point A, where AB is the diameter of a circle with centre C(2, −3)

C(2, −3) is the centre of the given circle. Let A(a, b) and B(1, 4) be the two end-points of the given diameter AB. Then, the coordinates of C are

$x=\frac{a+1}{2}, y=\frac{b+4}{2}$

It is given that $x=2$ and $y=-3$.

$\Rightarrow 2=\frac{a+1}{2},-3=\frac{b+4}{2}$

$\Rightarrow 4=a+1,-6=b+4$

$\Rightarrow a=4-1, b=-6-4$

$\Rightarrow a=3, b=-10$

Therefore, the coordinates of point A are (3, -10).