Question

the point (3,-4) is the centroid of circle if A B is a diameter of a circle and B is (5,-6) find the co ordinates of A​

Answer 1

Given:-

The coordinates of the center is (3,-4). Diameter AB where coordinates of B are (5,-6).

Required answer:-

The coordinates of points A.

Solution:-

Let the coordinates of A be (x,y)

We know, that AB is the diameter,

$\therefore\sf{OA = OB}$                                                                            $(\because{2r = d})$

$\sf{Since,\:O \:the\:midpoint \:of\:AB}$

$\therefore{ coordinates of O = }\sf{\frac{x_1+x_2}{2} ,\frac{y_1+y_2}{2}$, where, $\sf{x_1=5,x_2=x,y_1=-6\:and\:y_2=y}$

$\implies\sf{3,-4=}\sf{\frac{5+x}{2} ,\frac{-6+y}{2}$

Comparing the coordinates,

$\implies\sf{3=\frac{5+x}{2}$       and, $\sf{-4=\frac{-6+y}{2} }$

$\implies\sf{6=5+x}$     and, $\sf{-8=-6+y}$

$\implies\sf{x =1}$,    $\sf{y=-2}$

Hence, the coordinates of  A are (1,-2)