JEE Advanced · Mathematics · 14. Ellipse

Let $T_{1}$ and $T_{2}$ be two distinct common tangents to the ellipse $E : \frac{x^{2}}{6} + \frac{y^{2}}{3} = 1$ and the parabola $P : y^{2} = 12 x$ . Suppose that the tangent $T_{1}$ touches $P$ and $E$ at the points $A_{1}$ and $A_{2}$ , respectively and the tangent $T_{2}$ touches $P$ and $E$ at the points $A_{4}$ and $A_{3}$ , respectively. Then which of the following statements is(are) true?

A The area of the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ is $35$ square units
B The area of the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ is $36$ square units
C The tangents $T_{1}$ and $T_{2}$ meet the $x$ -axis at the point $(- 3, 0)$
D The tangents $T_{1}$ and $T_{2}$ meet the $x$ -axis at the point $(- 6, 0)$

Verified Solution

Answer & Solution

Correct Answer

(C) The tangents $T_{1}$ and $T_{2}$ meet the $x$ -axis at the point $(- 3, 0)$

Step-by-step Solution

Detailed explanation

Given, Equation of ellipse

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

	List- I		List- II
$(I)$	$2 h + k$	$(P)$	$6$
$(I I)$	$\frac{L e n g t h o f Z W}{L e n g t h o f X Y}$	$(Q)$	$\sqrt{6}$
$(I I I)$	$\frac{A r e a o f t r i a n g l e M Z N}{A r e a o f t r i a n g l e Z M W}$	$(R)$	$\frac{5}{4}$
$(I V)$	$α$	$(S)$	$\frac{21}{5}$
		$(T)$	$2 \sqrt{6}$
		$(U)$	$\frac{10}{3}$

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From JEE Advanced

An aqueous solution of X is added slowly to an aqueous solution of Y as shown in List I. The variation in conductivity of these reactions is given in List II. Match List I with List II and select the correct using answer the code given below the lists :

List - I	List - II
a. $\underset{\quad\quad~~ X }{( C _2 H _5)_3 N}+\underset{ Y }{CH _3 COOH}$	p. Conductivity decreases and then increases
b. $\underset{\quad\quad~~ X }{KI (0.1 M )}+\underset{ Y }{AgNO _3(0.01 M )}$	q. Conductivity decreases and then does not change much
c. $\underset{\text X }{CH _3 COOH}+\underset{\text Y }{KOH}$	r. Conductivity increases and then does not change much
d. $\underset{\text X }{NaOH}+\underset{\text Y }{HI}$	s. Conductivity does not change much and then increases

JEE Advanced 2013 Medium

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List - I	List - II
a. \(\underset{\quad\quad~~ X }{( C _2 H _5)_3 N}+\underset{ Y }{CH _3 COOH}\)	p. Conductivity decreases and then increases
b. \(\underset{\quad\quad~~ X }{KI (0.1 M )}+\underset{ Y }{AgNO _3(0.01 M )}\)	q. Conductivity decreases and then does not change much
c. \(\underset{\text X }{CH _3 COOH}+\underset{\text Y }{KOH}\)	r. Conductivity increases and then does not change much
d. \(\underset{\text X }{NaOH}+\underset{\text Y }{HI}\)	s. Conductivity does not change much and then increases