Let T 1 and T 2 be two distinct common tangents to the ellipse E : x 2 6 + 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 touche

Question

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 statement(s) is (are) true? [2023]

Smart Achievers · Accepted Answer

(1, 3)

$Given equation of ellipse E : \frac{x^{2}}{6} + \frac{y^{2}}{3} = 1 .$

$Tangent: y = m_{1} x \pm \sqrt{6 m_{1}^{2} + 3}$

$Equation of parabola P : y^{2} = 12 x$

$Tangent: y = m_{2} x + \frac{3}{m_{2}}$

$For common tangent$

$m_{1} = m_{2} = m (say) and \pm \sqrt{6 m_{1}^{2} + 3} = \frac{3}{m_{2}}$

$\Rightarrow m = \pm 1$

$Equation of common tangents are$

$T_{1} : y = x + 3 and T_{2} : y = - x - 3 .$

$∵$ $point of contact for parabola is (\frac{a}{m^{2}}, \frac{2 a}{m})$

$\Rightarrow A_{1} (3, 6) and A_{4} (3, - 6)$ $on solving y = x + 3 or y = - x - 3 and$

$equation of ellipse \frac{x^{2}}{6} + \frac{y^{2}}{3} = 1 we get A_{2} (- 2, 1) and A_{3} (- 2, - 1) .$

$Area of quadrilateral A_{1} A_{2} A_{3} A_{4} = \frac{1}{2} (12 + 2) \times 5$

$(∵ A_{1} A_{4} = 12, A_{2} A_{3} = 2, M N = 5)$

$= 35 sq. units.$

$Put y = 0 in T_{1} and T_{2} we get point of intersection with x-axis is (- 3, 0) .$

$Hence option (1) and (3) are correct.$

Solution

1 The area of the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ is 35 square units

2 The area of the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ is 36 square units

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

4 The tangents $T_{1}$ and $T_{2}$ meet the $x$ -axis at the point $(- 6, 0)$

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Solution

1 The area of the quadrilateral A1A2A3A4 is 35 square units

2 The area of the quadrilateral A1A2A3A4 is 36 square units

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

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

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1 The area of the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ is 35 square units

2 The area of the quadrilateral $A_{1} A_{2} A_{3} A_{4}$ is 36 square units

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

4 The tangents $T_{1}$ and $T_{2}$ meet the $x$ -axis at the point $(- 6, 0)$