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JEE Main202310 Apr 2023Evening ShiftMathematicsEllipseActual

Let a circle of radius 4 be concentric to the ellipse 15 x 2 + 19 y 2 = 285 . Then the common tangents are inclined to the minor axis of the ellipse at the angle

Options

  1. Aπ 3
  2. Bπ 4
  3. Cπ 6
  4. Dπ 12

Correct answer

A. π 3

Step-by-step solution

Given, 15 x 2 + 19 y 2 = 285 ⇒ x 2 19 + y 2 15 = 1 Now plotting the diagram with given circle of radius 4 we get, Now, let the equation of tangent of ellipse be y = m x ± 19 m 2 + 15 as   c 2 = a 2 m 2 + b 2 If it is tangent to circle x 2 + y 2 = 16 then using the formula, c 2 = r 2 1 + m 2 we get, 19 m 2 + 15 1 + m 2 = 4 ⇒ 19 m 2 + 15 = 16 + 16 m 2 ⇒ 3 m 2 = 1 ⇒ m = ± 1 3 ⇒ θ = 30 ° with x - axis, Hence, Angle made by tangent with minor axis i.e., with y axis

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