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JEE Main202127 Jul 2021Morning ShiftMathematicsEllipseActual

A ray of light through 2 , 1 is reflected at a point P on the y - axis and then passes through the point 5 , 3 . If this reflected ray is the directrix of an ellipse with eccentricity 1 3 and the distance of the nearer focus from this directrix is 8 53 , then the equation of the other directrix can be:

Options

  1. A11 x + 7 y + 8 = 0 or 11 x + 7 y - 15 = 0
  2. B11 x - 7 y - 8 = 0 or 11 x + 7 y + 15 = 0
  3. C2 x - 7 y + 29 = 0 or 2 x - 7 y - 7 = 0
  4. D2 x - 7 y - 39 = 0 or 2 x - 7 y - 7 = 0

Correct answer

C. 2 x - 7 y + 29 = 0 or 2 x - 7 y - 7 = 0

Step-by-step solution

The reflected point of 2 , 1 is - 2 , 1 . Equation of reflected ray y - 1 = 3 - 1 5 + 2 x + 2 ⇒ y - 1 = 2 7 x + 2 ⇒ 7 y - 7 = 2 x + 4 ⇒ 2 x - 7 y + 11 = 0 Let the equation of other directrix is 2 x - 7 y + λ = 0 Distance of directrix from focus a e - a e = 8 53 , where e is the eccentricity ⇒ 3 a - a 3 = 8 53 or a = 3 53 Distance from other focus a e + a e = 3 a + a 3 = 10 a 3 = 10 3 × 3 53 = 10 53 Distance between two directrix = 2 a e = 2 × 3 × 3 53 = 18 53 So, λ -

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