Tangent is drawn to ellipse (x^2)/27+(y^2)/1=1 at (3 √3 cos θ, sin θ) , where (θ ∈ (0, (π)/2 ) ) . Then, the value of θ such that the sum of intercept…

Question

Tangent is drawn to ellipse (x^2)/27+(y^2)/1=1 at (3 √3 cos θ, sin θ) , where (θ ∈ (0, (π)/2 ) ) . Then, the value of θ such that the sum of intercept on axes made by this tangent is minimum, is

MARKS Content Team · Accepted Answer

Correct answer: (B) (π)/6. Equation of tangent is drawn at a (3 √3 cos θ, sin θ) to the curve is (x cos θ)/(3 √3)+(y sin θ)/1=1 Thus, sum of intercepts =(3 √3 sec θ+cosec θ)=f(θ) (say) ⇒ f^′(θ)=3 √3 sec θ tan θ-cosec θ cot θ =(3 √3 sin^3 θ-cos^3 θ)/(sin^2 θ cos^2 θ) Put f^′(θ)=0 ⇒ sin^3 θ=frac13^3 / 2 cos^3 θ ⇒ tan θ=1/(√3) ⇒ θ=(π)/6

Tangent is drawn to ellipse $\frac{x^2}{27}+\frac{y^2}{1}=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$, where $\left(\theta \in\left(0, \frac{\pi}{2}\right)\right)$. Then, the value of $\theta$ such that the sum of intercept on axes made by this tangent is minimum, is

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