If a tangent of slope 2 of the ellipse a 2 x 2 + b 2 y 2 = 1 is normal to the circle x 2 + y 2 + 4 x + 1 = 0 , then the maximum value of ab is ____

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If a tangent of slope 2 of the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ is normal to the circle $x^{2} + y^{2} + 4 x + 1 = 0$ , then the maximum value of ab is $____$

Answer:

Solution:

A tangent of slope 2 is $y = 2 x \pm 4 a^{2} + b^{2} \to (1)$

This is normal to the circle $x^{2} + y^{2} + 4 x + 1 = 0$ i.e.,

(1) passes through $(- 2, 0) 4 a^{2} + b^{2} = 16$ Using $A M \geq GM \Rightarrow \frac{4 a ^{2} + b ^{2}}{2} \geq 4 a^{2} \cdot b^{2} ab \leq 4$

Stream:JEESubject:MathematicsTopic:EllipseSubtopic:Tangent to ellipse

⏱ 2mℹ️ Source: QB

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