Problem
Let T₁ and T₂ be two distinct common tangents to the ellipse E: x²/6 + y²/3 = 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)
Solution
Any tangent to the parabola is of the form y = mx + 3/m. We have to find the value of m such that this line is also a tangent to the ellipse.
One way to do that is to substitute the value of y in the ellipse and equate the discriminant of the resulting quadratic equation to 0.
Another way is to consider any tangent to the ellipse, which of the form y = mx ± √(6m² + 3).
Since these are the same tangents, we have 3/m = ± √(6m² + 3). This gives the equation 2m⁴ + m² – 3 = 0, which gives m = ± 1.
So, the two common tangents are y = x + 3 and y = -x – 3, which meet at (-3, 0). This makes option C true, and option D false.
Now, to find the area of the quadrilateral, let’s draw things out.
The area can be calculated by subtracting the area of ΔABA₂ from ΔACA₁, and doubling the answer.
Now, A₂A₃ is the chord of contact of the ellipse from the point A, whose equation can be found using the equation T = 0, or x(-3)/6 + y(0)/3 = 1. This simplifies to x = -2.
This means that B is (-2, 0), and AB = 1 = BA₂ (since ∠A₂AB = 45°). So, the area of ΔABA₂ equals 1/2 x 1 x 1 or 1/2.
Similarly, A₁A₄ is the chord of contact of the parabola from the point A, whose equation is y(0) = 6(x – 3) or x = 3.
This means that C is (3, 0), and AC = 6 = CA₁ (since ∠A₁AC = 45°). So, the area of ΔACA₁ equals 1/2 x 6 x 6 or 18.
Finally, the required area is 2(ar(ΔACA₁) – ar(ΔABA₂)) = 2(18 – 1/2) = 35 square units. Therefore, option A is true and option B is false.
Comments
A fairly easy / standard problem based on common tangents, involving a fair bit of calculations. Boring.