Question
If $$P = \left( {1,\,0} \right),\,Q = \left( { - 1,\,0} \right)$$ and $$R = \left( {2,\,0} \right)$$ are three given points, then locus of the point $$S$$ 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 :
a straight line parallel to $$x$$-axis
Solution :
$$\eqalign{
& {\text{We have}} \cr
& P = \left( {1,\,0} \right),\,Q = \left( { - 1,\,0} \right),\,R = \left( {2,\,0} \right) \cr
& {\text{Let}}\,\,\,{\text{ }}S = \left( {x,\,y} \right) \cr
& {\text{ATQ}}\,\,\,S{Q^2} + S{R^2} = 2S{P^2} \cr
& \Rightarrow {\left( {x + 1} \right)^2} + {y^2} + {\left( {x - 2} \right)^2} + {y^2} = 2\left[ {{{\left( {x - 1} \right)}^2} + {y^2}} \right] \cr
& \Rightarrow 2{x^2} + 2{y^2} - 2x + 5 = 2{x^2} + 2{y^2} - 4x + 2 \cr
& \Rightarrow 2x + 3 = 0 \cr
& \Rightarrow x = - \frac{3}{2} \cr} $$
Which is a straight line parallel to $$y$$-axis.