JEE Advanced · Mathematics · 14. Ellipse

Let $E_{1}$ and $E_{2}$ be two ellipse whose centers are at the origin. The major axes of $E_{1} a n d E_{2}$ lie along the x - axis and the y - axis, respectively. Let S be the circle $x^{2} + {(y - 1)}^{2} = 2 .$ The straight line $x + y = 3$ touches the curves $S, E_{1}$ and $E_{2}$ at $P, Q$ and $R,$ respectively. Suppose that $P Q = P R = \frac{2 \sqrt{2}}{3} .$ If $e_{1}$ and $e_{2}$ are the eccentricities of $E_{1}$ and $E_{2},$ respectively, then the correct expression(s) is(are)

A $e_{1}^{2} + e_{2}^{2} = \frac{43}{40}$
B $e_{1} e_{2} = \frac{\sqrt{7}}{2 \sqrt{10}}$
C $|e_{1}^{2} - e_{2}^{2}| = \frac{5}{8}$
D $e_{1} e_{2} = \frac{\sqrt{3}}{4}$

Verified Solution

Answer & Solution

Correct Answer

(B) $e_{1} e_{2} = \frac{\sqrt{7}}{2 \sqrt{10}}$

Step-by-step Solution

Detailed explanation

Let Q be

(x_{1}, y_{1})

So equation tangent at Q will be

x x_{1} + y y_{1} - (y + y_{1}) - 1 = 0

Comparing with

x + y = 3

(x_{1} y_{1}) \equiv (1,2)

So R and Q will be

(1 \pm \frac{2 \sqrt{2}}{3} \cos \frac{3 π}{4}, 2 \pm \frac{2 \sqrt{2}}{3} \sin \frac{3 π}{4})

\Rightarrow Q (\frac{5}{3}, \frac{4}{3}) a n d (\frac{1}{3}, \frac{8}{3})

Let Q lies on

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

So tangent at P is

\frac{5 x}{3 a^{2}} + \frac{4 y}{3 b^{2}} = 1

Comparing with

x + y = 3

a^{2} = 5, b^{2} = 4 \Rightarrow e_{1} = \frac{1}{\sqrt{5}}

And R lies on

\frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{b_{1}^{2}} = 1

So tangent at

\frac{x}{{3 a}_{1}^{2}} + \frac{8 y}{{3 b}_{1}^{2}} = 3

Comparing with

x + y = 3

\Rightarrow a_{1}^{2} = 1, b_{1}^{2} = 8 \Rightarrow e_{2} = \sqrt{\frac{7}{8}}

\Rightarrow e_{1}^{2} + e_{2}^{2} = \frac{43}{40} a n d e_{1} e_{2} = \frac{\sqrt{7}}{2 \sqrt{10}}

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.

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From JEE Advanced

Let

p, q, r

be non-zero real numbers that are, respectively, the

10^{th}, 100^{th}

and

1000^{th}

terms of a harmonic progression. Consider the system of linear equations

x + y + z = 1

10 x + 100 y + 1000 z = 0

q r x + p r y + p q z = 0

	List- $I$		List- $II$
$(I)$	If $\frac{q}{r} = 10$ , then the system of linear equations has	$(P)$	$x = 0, y = \frac{10}{9}, z = - \frac{1}{9}$ as a solution
$(II)$	If $\frac{p}{r} \neq 100$ , then the system of linear equations has	$(Q)$	$x = \frac{10}{9}, y = - \frac{1}{9}, z = 0$ as a solution
$(III)$	If $\frac{p}{q} \neq 10$ , then the system of linear equations has	$(R)$	infinitely many solutions
$(IV)$	If $\frac{p}{q} = 10$ , then the system of linear equations has	$(S)$	no solution
		$(T)$	at least one solution

The correct option is:

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