Let $ L $ be a common tangent line to the curves $ 4 x^{2}+9 y^{2}=36 $ and $ (2 x)^{2}+(2 y)^{2}=31 . $ Then the square of the slope of the line $ L $ is

Question

Let $ L $ be a common tangent line to the curves $ 4 x^{2}+9 y^{2}=36 $ and $ (2 x)^{2}+(2 y)^{2}=31 . $ Then the square of the slope of the line $ L $ is,

MARKS App · Accepted Answer

Given curves are x 2 9 + y 2 4 = 1 x 2 + y 2 = 31 4 let slope of common tangent be m so tangents are y = m x ± 9 m 2 + 4 y = m x ± 31 2 1 + m 2 hence 9 m 2 + 4 = 31 4 1 + m 2 ⇒ 36 m 2 + 16 = 31 + 31 m 2 ⇒ m 2 = 3

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