The tangent at point (a cos θ, b sin θ), 0 < θ < (π)/2 , to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 meets the x -axis at T and y -axis at T_1 . Then the value of min _0 < θ < (π)/2(O T) (O T_1 ) is

Correct answer: (B) 2 ab. Hint: Tangent : x(b cos θ)+y(a sin θ)=a b ∴ OT · OT_1=ab/(cos θ sin θ)=(2 ab)/(sin 2 θ)

Q84 / 155 · WBJEE 2023 · Mathematics

The tangent at point \((a \cos \theta, b \sin \theta), 0 < \theta < \frac{\pi}{2}\), to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) meets the \(x\)-axis at \(T\) and \(y\)-axis at \(T_1\). Then the value of \(\min _{0 < \theta < \frac{\pi}{2}}(O T)\left(O T_1\right)\) is

A
\(a b\)
Correct answer
B
\(2 \mathrm{ab}\)
C
0
D
1

Solution

Hint: Tangent : \(x(b \cos \theta)+y(a \sin \theta)=a b\) \(\therefore \mathrm{OT} \cdot \mathrm{OT}_1=\frac{\mathrm{ab}}{\cos \theta \sin \theta}=\frac{2 \mathrm{ab}}{\sin 2 \theta}\)

Hence, the correct answer is option: 'B'.

The tangent at point \((a \cos \theta, b \sin \theta), 0 < \theta < \frac{\pi}{2}\), to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) meets the \(x\)-axis at \(T\) and \(y\)-axis at \(T_1\). Then the value of \(\min _{0 < \theta < \frac{\pi}{2}}(O T)\left(O T_1\right)\) is

Solution

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