JEE Advanced · Mathematics · 14. Ellipse

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle \(R\) whose sides are parallel to the coordinate axes. Another ellipse \(E_{2}\) passing through the point \((0,4)\) circumscribes the rectangle \(R\). The eccentricity of the ellipse \(E_{2}\) is

A \(\frac{\sqrt{2}}{2}\)
B \(\frac{\sqrt{3}}{2}\)
C \(\frac{1}{2}\)
D \(\frac{3}{4}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{1}{2}\)

Step-by-step Solution

Detailed explanation

As rectangle \(A B C D\) circumscribed the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1, \quad \therefore A=(3,2)\)

Let the ellipse circumscribing the rectangle \(A B C D\) is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1...(i)\)

Given that ellipse (i) passes through \((a, 4)\) \(\therefore b^{2}=16\)

Also ellipse (i) passes through \(A(3,2)\)

\(\therefore \quad \frac{9}{a^{2}}+\frac{4}{16}=1 \Rightarrow a^{2}=12\)

\(\therefore \quad e=\sqrt{1-\frac{12}{16}}=\sqrt{\frac{1}{4}}=\frac{1}{2}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Paragraph:

Let \(O\) be the origin, and \(\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}\) be three unit vectors in the directions of the sides \(\overrightarrow{Q R}, \overrightarrow{R P}\), \(\overrightarrow{P Q}\), respectively, of a triangle \(P Q R\).

Question:

\(|\overrightarrow{O X} \times \overrightarrow{O Y}|=\)

JEE Advanced 2017 Easy

Let \(\mathbb{R}\) denote the set of all real numbers. For a real number \(x\), let \([x]\) denote the greatest integer less than or equal to \(x\). Let \(n\) denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	The minimum value of \(n\) for which the function \(f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]\) is continuous on the interval \([1,2]\),is	(1)	8
(Q)	The minimum value of \(n\) for which \(g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}\),is an increasing function on \(\mathbb{R}\),is	(2)	9
(R)	The smallest natural number \(n\) which is greater than 5,such that \(x=3\) is a point of local minima of \(h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)\),is	(3)	5
(S)	Number of \(x_0 \in \mathbb{R}\) such that \(l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}\),is NOT differentiable at \(x_0\),is	(4)	6
		(5)	10

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