If , and are three given points, then locus of the points satisfying the relation 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
Show answer and solution
Correct option: D
Solution
Let S = (x, y). Compute each distance squared:
SQ² = (x+1)² + y²
SR² = (x-2)² + y²
SP² = (x-1)² + y²
The condition SQ² + SR² = 2SP² gives:
(x+1)² + y² + (x-2)² + y² = 2[(x-1)² + y²]
Expand: x²+2x+1 + x²-4x+4 + 2y² = 2x²-4x+2 + 2y²
2x² - 2x + 5 + 2y² = 2x² - 4x + 2 + 2y²
-2x + 5 = -4x + 2
2x = -3
x = -3/2
This is a straight line parallel to the Y-axis.
SQ² = (x+1)² + y²
SR² = (x-2)² + y²
SP² = (x-1)² + y²
The condition SQ² + SR² = 2SP² gives:
(x+1)² + y² + (x-2)² + y² = 2[(x-1)² + y²]
Expand: x²+2x+1 + x²-4x+4 + 2y² = 2x²-4x+2 + 2y²
2x² - 2x + 5 + 2y² = 2x² - 4x + 2 + 2y²
-2x + 5 = -4x + 2
2x = -3
x = -3/2
This is a straight line parallel to the Y-axis.
[1988]