Question

If P(2, – 7) is given point and Q is a point on 2x² + 9y² = 18 then find the equations of the locus of the midpoint of PQ

Answer 1

Given P is (2, -7) and let Q be (x, y)

Given that Q is a point on 2x² + 9y² = 18

Let M (h, k) be the midpoint of PQ

∴ (h, k) = `((2 + x)/2, - (7 + y)/2)`

h = `(2 + x)/2`

k = `-(7 + y)/2`

2h = 2 + x ,

2k = – 7 + y

x = 2h – 2,

y = 2k + 7

But Q(x, y) is a point on 2x² + 9y² = 18

∴ 2(2h – 2)² + 9(2k + 7)² = 18

2[4h² – 8h + 4] + 9[4k² + 28k + 49] = 18

8h² – 16h + 8 + 36k² + 252k + 441 = 18

8h² + 36k² – 16h + 252k + 449 = 18

8h² + 36k² – 16h + 252k + 431 =0

The locus of M(h, k) is obtained by replacing h by x and k by y.

∴ The required locus is 8x² + 36y² – 16x + 252y + 431 = 0