Class 10

Math

All topics

Coordinate Geometry

Find the coordinates of the point equidistant from three given points $A(5,3),B(5,−5)$ and $C(1,−5)$.

Let the coordintaes of the point be $O(x,y)$, then

$OA=OB=OC$

or $OA_{2}=OB_{2}=OC_{2}$

$OA_{2}=(5−x)_{2}+(3−y)_{2}$

$OB_{2}=(5−x)_{2}+(−5−y)_{2}$

$OC_{2}=(1−x)_{2}+(−5−y)_{2}$

$(5−x)_{2}+(3−y)_{2}=(5−x)_{2}+(−5−y)_{2}$

$9−6y+y_{2}=25+10y+y_{2}$

$9−25=10y+6y$

$⇒16y=−16$

$y=16−16 =−1$

and $(5−x)_{2}+(−5−y)_{2}$

$=(1−x)_{2}+(−5−y)_{2}$

$25−10x+x_{2}=1−2x+x_{2}$

$−10x+2x=1−25$

$−8x=−24$

or $x=3$

So, coordinates of the point is $(3,1)$.