Find the coordinates of a point $A$, where $AB$ is the diameter of a circle whose centre is $(2, -3)$ and $B$ is $(1, 4)$.

Given:

AB is diameter of a circle whose centre is $( 2,\ -3)$ and B is $( 1,\ 4)$.

To do:

We have to find the coordinates of point A.

Solution:

Let the centre be $O(2,\ -3)$ and coordinates of point A be $( x,\ y)$.

$AB$ is the diameter of the circle with centre $O$.

This implies,

$O$ is the mid-point of AB.

We know that,

Mid-point of two points $( x_{1},\ y_{1})$ and $( x_{2},\ y_{2})$ is,

$(x,y)=( \frac{x_{1}+x_{2}}{2},\ \frac{y_{1}+y_{2}}{2})$

Using the mid-point formula,

$( 2,\ -3)=( \frac{x+1}{2},\  \frac{y+4}{2})$

Equating the coordinates on both sides, we get,

$\frac{x+1}{2}=2$ and $\frac{y+4}{2}=-3$

$\Rightarrow x+1=2(2)$ and $y+4=-3(2)$

$\Rightarrow x+1=4$ and $y+4=-6$

$\Rightarrow x=4-1$  and $y=-6-4$

$\Rightarrow x=3$ and $y=-10$

The coordinates of point A are $(3,-10)$.