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JEE Main202226 Jun 2022Morning ShiftMathematicsParabolaActual

Let the common tangents to the curves 4 x 2 + y 2 = 9 and y 2 = 4 x intersect at the point Q . Let an ellipse, centered at the origin O , has lengths of semi-minor and semi-major axes equal to O Q and 6 , respectively. If e and l respectively denote the eccentricity and the length of the latus rectum of this ellipse, then l e 2 is equal to ______.

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Step-by-step solution

Given curves are x 2 + y 2 = 9 4   &   y 2 = 4 x Equation of tangent in slope form to the circle and parabola will be y = m x ± 3 2 1 + m 2           … i y = m x + 1   m         ⋯ i i Comparing i & i i , we get ± 3 2 1 + m 2 = 1 m 9 m 2 1 + m 2 = 4 9 m 4 + 9 m 2 - 4 = 0 9 m 4 + 12 m 2 - 3 m 2 - 4 = 0 3 m 2 3 m 2 + 4 - 3 m 2 + 4 = 0 here m 2 = - 4 3 (Rejected) so m 2 = 1 3   ⇒ m = ± 1 3 So, the equation of common ta

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