Let T1 and T2 be two distinct common tangents to the ellipse E : x^2 /6 + y^2 /3 and the parabola P : y^2 = 12x. ​

Question

Let T₁ and T₂ be two distinct common tangents to the ellipse E : \(\frac{x^2}{6} + \frac{y^2}3 = 1\) and the parabola P : y² = 12x. Suppose that the tangent T₁ touches P and E at the points A₁ and A₂, respectively and the tangent T₂ touches P and E at the points A₄ and A₃, respectively. Then which of the following statements is(are) true?

(A) The area of the quadrilateral A₁A₂A₃A₄ is 35 square units

(B) The area of the quadrilateral A₁A₂A₃A₄ is 36 square units

(C) The tangents T₁ and T₂ meet the x-axis at the point (–3, 0)

(D) The tangents T₁ and T₂ meet the x-axis at the point (–6, 0)

Answer 1

Correct options are (A) & (C)

Equation of common tangents y = x + 3 and y = –x – 3 point of contact for parabola is \(\left(\frac a{m^2}, \frac{2a}3\right)\)

A₁ ≡ (3, 6), A₄(3 - 6)

Let A₂(x₁, y₁)

⇒ tangent to E is \(\frac{xx_1}6 + \frac{yy_1}3 = 1\)

A₃ is mirror image of A₂ in x-axis ⇒ A₃(–2, –1)

Intersection point of T₁ = 0 and T₂ = 0 is (–3, 0)

Area of quadrilateral A₁A₂A₃A₄ = \(\frac 12\) (12 + 2) x 5 = 35 square units.