# If the coordinates of the one end of a diameter of a circle are $( 2,\ 3)$ and the coordinates if its center are $(-2,\ 5)$, then the coordinates of the other end of the diameter are:$( A)( -6,\ 7)$ $( B)( 6,\ -7)$ $( C)( 6,\ 7)$ $( D)( -6,\ -7)$

Given: A circle with the center $( -2,\ 5)$ and one end of its diameter $( 2,\ 3)$.

To do: To find out the co-ordinates of the other end of the diameter of the given circle.

Solution: Let there is a circle with the center O and its diameter is AB.

Here as given $O=( -2,\ 5)$ and $A=( 2,\ 3)$ and $B=?$

Here it is clear if AB is the diameter of the given circle with center O, then O is the mid-point of AB.

Mid-point $P( x,\ y) =\left(\frac{x_{1} +x_{2}}{2} ,\frac{y_{1} +y_{2}}{2}\right)$

Here $-2,\ 5=\left(\frac{2+x_{2}}{2} ,\frac{3+y_{2}}{2}\right)$

$\Rightarrow -2=\frac{2+x_{2}}{2} \ and\ 5=\frac{3+y_{2}}{2}$

$\Rightarrow 2+x_{2} =-4\ and\ 3+y_{2} =10$

$\Rightarrow x_{2}=-6\ and\ y_{2}=7$

$\therefore \$Co-ordinates of the other end of the diameter of the given circle are $( -6,\ 7)$
$\therefore$ Option $( A)$ is correct.