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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
Solution
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
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