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The length of the portion of the common tangent to x 2 + y 2 = 16 and 9 x 2 + 25 y 2 = 225 between the two points of contact is

Options

  1. A9 4 units
  2. B3 4 units
  3. C3 4 7 units
  4. D5 4 7 units

Correct answer

C. 3 4 7 units

Step-by-step solution

Equation of the ellipse is x 2 25 + y 2 9 = 1 Any point on the ellipse is 5 cos ⁡ θ , 3 sin ⁡ θ at which the equation of the tangent is x 5 cos ⁡ θ + y 3 sin ⁡ θ = 1 If it is also a tangent to the circle x 2 + y 2 = 16 , then its distance from the origin is equal to the radius, ⇒ 1 c o s 2 θ 25 + s i n 2 θ 9 = 4 ⇒ cos 2 θ 25 + 1 - cos 2 θ 9 = 1 16 ⇒ cos ⁡ θ = 5 7 16 If l be the length of the tangent then l 2 = S ' or l 2 = 25 c o s 2 θ + 9 s i n 2 θ - 16 = 16 c o s 2 θ - 7 = 16 . 25.7 16 2 - 7 = 175 - 112 16 = 63 1

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