JEE Advanced 2023 · Paper 1 · Mathematics Q2

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MathematicsMultiple correct+4 / -2Question 2 of 51

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E : \dfrac{x^2}{6} + \dfrac{y^2}{3} = 1$ and the parabola $P : y^2 = 12x$ . 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?

The area of the quadrilateral

A_1A_2A_3A_4

is 35 square units

The area of the quadrilateral

A_1A_2A_3A_4

is 36 square units

The tangents

T_1

and

T_2

meet the

x

-axis at the point

(-3, 0)

The tangents

T_1

and

T_2

meet the

x

-axis at the point

(-6, 0)

Answer and worked solution

Official answer: A, C

Common tangents. A tangent to $y^2=12x$ is $y=mx+\dfrac3m$ . It touches $\dfrac{x^2}{6}+\dfrac{y^2}{3}=1$ when $c^2=6m^2+3$ :
$\frac9{m^2}=6m^2+3\implies 2m^4+m^2-3=0\implies m^2=1.$
So $T_1: y=x+3$ and $T_2: y=-x-3$ . Both cut the $x$ -axis at $(-3,0)$ , so (C) is true and (D) is false.

Points of contact.

On the parabola, $\left(\frac{a}{m^2},\frac{2a}{m}\right)$ with $a=3$ : $A_1=(3,6)$ and $A_4=(3,-6)$ .
On the ellipse, $\left(-\frac{6m}{c},\frac{3}{c}\right)$ : $A_2=(-2,1)$ and $A_3=(-2,-1)$ .

Area. $A_1A_2A_3A_4$ is a trapezium with vertical parallel sides 12 and 2, a distance 5 apart:
$\text{Area}=\tfrac12(12+2)\cdot5=35.$
So (A) is true and (B) is false.

Topics:Tangents and normals to conics Ellipse Parabola

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