If a circle and ellipse is given by the equations $ x^{2}+y^{2}=16 $ and $ \frac{x^{2}}{25}+\frac{y^{2}}{4}=1 $, then the equation of common tangent between circle and ellipse in the first quadrant is

Question

If a circle and ellipse is given by the equations $ x^{2}+y^{2}=16 $ and $ \frac{x^{2}}{25}+\frac{y^{2}}{4}=1 $, then the equation of common tangent between circle and ellipse in the first quadrant is, $ y=-\frac{2 x}{\sqrt{3}}+4 \sqrt{\frac{7}{3}} $, $ y=\frac{2 x}{\sqrt{3}}+4 \sqrt{\frac{7}{3}} $, $ y=\frac{-2 x}{\sqrt{3}}-4 \sqrt{\frac{7}{3}} $, None of these

MARKS App · Accepted Answer

Let equation of tangent to x 2 + y 2 = 1 6 and x 2 2 5 + y 2 4 = 1 be simultaneously y = mx + 4 1 + m 2 . . . ( i ) and y = mx + 2 5 m 2 + 4 . . . ( ii ) Since, equations ( i ) and ( ii ) are same tangent. ∴ 4 1 + m 2 = 2 5 m 2 + 4 ⇒ 1 6 1 + m 2 = 2 5 m 2 + 4 ⇒ 9 m 2 = 1 2 ⇒ m = ± 2 3 As m 0 ⇒ m = - 2 3 So, the equation of the common tangent is, y = - 2 x 3 + 4 7 3 .

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