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The product of the slopes of the common tangents of the ellipse x 2 + 4 y 2 = 16 and the parabola y 2 - 4 x - 4 = 0 is

Options

  1. A- 1 15
  2. B- 1 5
  3. C- 1 3
  4. D- 1 2

Correct answer

B. - 1 5

Step-by-step solution

Given, ellipse: x 2 16 + y 2 4 = 1 and parabola: y 2 = 4 x + 1 Putting x + 1 = X and Y = y , we get, Y 2 = 4 X So, the equation of the tangent is Y = m X + 1 m = m x + 1 + 1 m Y = m x + m + 1 m … i If i is also a tangent to x 2 16 + y 2 4 = 1 , then, c 2 = a 2 m 2 + b 2 ⇒ m + 1 m 2 = 16 m 2 + 4 ⇒ m 2 + 1 m 2 + 2 = 16 m 2 + 4 ⇒ 15 m 2 - 1 m 2 + 2 = 0 Let, m 2 = t > 0 ⇒ 15 t 2 + 2 t - 1 = 0 ⇒ t = 1 5 , - 1 3 ⇒ m 2 = 1 5 Hence, m = ± 1 5

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