Let the points P and Q lie on the lines y = x and y = 2 x respectively. The equation of the locus of the mid-point of the line segment joining P and Q , if P Q = 4 , is

Question

Let the points P and Q lie on the lines y = x and y = 2 x respectively. The equation of the locus of the mid-point of the line segment joining P and Q , if P Q = 4 , is, 25 x 2 + 36 x y + 13 y 2 = 4, 25 x 2 - 36 x y + 13 y 2 = 4, 25 x 2 - 36 x y - 13 y 2 = 4, 25 x 2 + 36 x y - 13 y 2 = 4

MARKS App · Accepted Answer

Let, P a , a and Q b , 2 b and the midpoint of P Q is h , k 2 h = a + b 2 k = a + 2 b a = 2 2 h - k and b = 2 k - h Now, a - b 2 + a - 2 b 2 = 16 ⇒ 6 h - 4 k 2 + 8 h - 6 k 2 = 16 ⇒ 9 h 2 + 4 k 2 - 12 h k + 16 h 2 + 9 k 2 - 24 h k = 4 ⇒ 25 x 2 + 13 y 2 - 36 x y = 4

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