Question

If $P=(1,0);Q=(-1,0)$ & $R=(2,0)$ are three given points, then the locus of the points S satisfying the relation, $S{Q}^2+S{R}^2=2S{P}^2$ is

• A. A straight line parallel to x-axis
• B. A circle passing through the origin
• C. A circle with the centre at the origin
• D. A straight line parallel to y-axis

Answer 1

The correct option is D A straight line parallel to y-axis

Let $S=(x,y)$
Then
$S{R}^2=(x-2{)}^2+y^2$
$S{Q}^2=(x+1{)}^2+y^2$
$S{P}^2=(x-1{)}^2+y^2$
Now
$S{Q}^2+S{R}^2=2S{P}^2$ implies
$(x-2{)}^2+y^2+(x+1{)}^2+y^2=2\left(\right(x-1{)}^2+y^2)$
$2x^2+2y^2-2x+5=2x^2-4x+2+2y^2$
$-2x+5=-4x+2$
$2x=-3$
$x=-3/2$
This is the equation of a line parallel to y axis.