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

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

What to do: To Find the coordinates of a point A.

Solution:

Let the centre be O and co-ordinates of point A be $( x,\ y)$

$\because AB is the diameter of Circle with centre O.

$\therefore O is the mid-point of AB.

As known if we have two points $( x_{1},\ y_{1})$ and $( x_{2},\ y_{2})$

Then mid-point $=( \frac{x_{1}+x_{2}}{2},\ \frac{y_{1}+y_{2}}{2})$

Using the mid-point formula,

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

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

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

$\Rightarrow x=4-1$ and $y=-3-4$

$\Rightarrow x=3$ and $y=-7$

Hence, Co-ordinates of $A=( 3,\ -7)$

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