QJEEAdv23Ma53d3: JEE Advanced 2023 Mathematics - Class 11 JEEAdv23Ma53d3 Solution: Let and be two distinct common tangents to the ellipse and the…

Question

Verified Solution • 2025–26 Session moderate

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?

Accepted Answer

\begin{aligned} E : \frac{x^{2}}{6} + \frac{y^{2}}{3} = 1, Tangent : y = m_{1} x \pm \sqrt{6 m_{1}^{2} + 3} \\ P : y^{2} = 12 x, Tangent: y = m_{2} x + \frac{3}{m_{2}} \end{aligned}

For common tangent

\begin{aligned} m = m_{1} = m_{2}, \pm \sqrt{6 m_{1}^{2} + 3} = \frac{3}{m_{2}} \\ \Rightarrow m = \pm 1 \end{aligned}

\Rightarrow

Equation of common tangents

y=x+3

and

y=-x-3

point of contact for parabola is

\left(\frac{a}{m^2}, \frac{2 a}{m}\right)

\Rightarrow A_{1} \equiv (3, 6), A_{4} (3 - 6)

Let

A_2\left(x_1, y_1\right) \Rightarrow

tangent to

E

is

\frac{x x_1}{6}+\frac{y y_1}{3}=1

A_3

is mirror image of

A_2

in

x

-axis

\Rightarrow A_3(-2,-1)

Intersection point of

T_1=0

and

T_2=0

is

(-3,0)

Area of quadrilateral

A_1 A_2 A_3 A_4=\frac{1}{2}(12+2) \times 5=35

square units

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