If a tangent of slope 13 of the ellipse x2a2+y2b2=1(a>b>0) is normal to the circle x2+y2+2x+2y+1=0 then which of the following is/are CORRECT?

maximum value of ab is 2 3 , a ∈ 2 5 , 2

If a tangent of slope 13 of the ellipse x2a2+y2b2=1(a>b>0) is normal to the circle x2+y2+2x+2y+1=0 then which of the following is/are CORRECT?

Equation of tangent y = 1 3 x ± a 2 9 + b 2 it passes through (-1, -1) − 3 = − 1 ± a 2 + 9 b 2 a 2 + 9 b 2 = 4 AM ≥ GM 4 2 ≥ a .3 b ≤ 2 3 a 2 + 9 a 2 1 − e 2 = 4 ⇒ e 2 = 10 a 2 − 4 9 a 2 0 < e 2 < 1 ⇒ a ∈ 2 5 , 2

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If a tangent of slope $\frac{1}{3}$ of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ is normal to the circle $x^{2} + y^{2} + 2 x + 2 y + 1 = 0$ then which of the following is/are CORRECT?

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maximum value of ab is $\frac{2}{3}$

$a \in (\sqrt{\frac{2}{5}}, 2)$

$a \in (\frac{2}{3}, 2)$

maximum value of ab is 1

answer is A, B.

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Detailed Solution

Equation of tangent $y = \frac{1}{3} x \pm \sqrt{\frac{a^{2}}{9} + b^{2}}$ it passes through (-1, -1)
$\begin{array}{l} - 3 = - 1 \pm \sqrt{a^{2} + 9 b^{2}} \\ a^{2} + 9 b^{2} = 4 \\ AM \geq GM \frac{4}{2} \geq a .3 b \leq \frac{2}{3} \\ a^{2} + 9 a^{2} (1 - e^{2}) = 4 \Rightarrow e^{2} = \frac{10 a^{2} - 4}{9 a^{2}} \\ 0 < e^{2} < 1 \Rightarrow a \in (\sqrt{\frac{2}{5}, 2}) \end{array}$

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