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JEE Main202430 Jan 2024Evening ShiftMathematicsStraight LinesActual

If x 2 - y 2 + 2 h x y + 2 g x + 2 f y + c = 0 is the locus of a point, which moves such that it is always equidistant from the lines x + 2 y + 7 = 0 and 2 x - y + 8 = 0 , then the value of g + c + h - f equals

Options

  1. A14
  2. B6
  3. C8
  4. D29

Correct answer

A. 14

Step-by-step solution

Let P ( x , y ) be the point whose distance from x + 2 y + 7 = 0 and 2 x - y + 8 = 0 is equal. ⇒ x + 2 y + 7 5 = ± 2 x - y + 8 5 ⇒ x 2 + 4 y 2 + 49 + 4 x y + 28 y + 14 x = 4 x 2 + y 2 + 64 - 4 x y - 16 y + 32 x ⇒ - 3 x 2 + 3 y 2 - 15 + 8 x y + 44 y - 18 x = 0 ⇒ 3 x 2 - 3 y 2 - 8 x y + 18 x - 44 y + 15 = 0 ⇒ x 2 - y 2 - 8 3 x y + 6 x - 44 3 y + 5 = 0 Now, on comparing x 2 - y 2 + 2 h x y + 2 g x + 2 f y + c = 0 with above equation we get, ⇒ h = - 4 3 , g = 3 , f = - 22 3 , c = 5 ⇒ g + c + h - f = 3 + 5 - 4 3 + 22 3

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