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Mathematics Question on Straight lines

If P=(1,0),Q=(−1,0)P = (1,0), Q = (-1,0) and R=(2,0)R= (2,0) are three given points, then locus of the points satisfying the relation SQ2+SR2=2SP2,SQ^2 + SR^2 = 2SP^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 7-axis

Answer

a straight line parallel to 7-axis

Explanation

Solution

The correct option is(D): a straight line parallel to 7-axis.

Let the coordinate of S be (x, y)
\because \hspace15mm SQ^2 + SR^2 = 2SP^2
⇒(x+1)2+y2+(x−2)2+y2=2[(x−1)2+y2]\Rightarrow (x + 1)^2 + y^2 + (x - 2)^2 + y^2 = 2 [(x - 1)^2 + y^2]
⇒x2+2x+1+y2+x2−4x+4+y2=2(x2−2x+1+y2)\Rightarrow x^2+ 2x+ 1 + y^2+ x^2-4x+ 4 + y^2=2(x^2- 2 x + 1 + y^2)
⇒2x+3=0⇒x=−32\Rightarrow 2 x + 3 = 0 \Rightarrow x=- \frac{3}{2}
Hence, it is a straight line parallel to 7-axis.