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 touches P and E at the points A 4 and A 3 , respectively. Then which of the following statements is(are) true?
C. The tangents T 1 and T 2 meet the x -axis at the point – 3 ,   0
Given, Equation of ellipse E : x 2 6 + y 2 3 = 1 , Now equation of tangent with slope m 1 will be : T 1 :   y = m 1 x ± 6 m 1 2 + 3 And equation of parabola, P : y 2 = 12 x , So, equation of tangent with slope m 2 will be: y = m 2 x + 3 m 2 Now for common tangent m = m 1 = m 2 ,   ± 6 m 1 2 + 3 = 3 m 2 ⇒   m = ± 1 Hence, equation of common tangents will be, y = x + 3 and y = – x - 3 Now we know that, Point of contact for parabola is a m 2 ,   2 a m ⇒   A 1 ≡ 3 ,   6 ,   A 4 3 - 6 Now let A 2 x 1 ,   y 1 So, equation of tangent to ellipse E at point A 2 x 1 ,   y 1 is given by, x x 1 6 + y y 1 3 = 1 Now comparing above equation with y = x + 3 , We get, - x 1 6 = y 1 3 = 1 3 ⇒ A 2 ≡ - 6 3 , 3 3 ≡ - 2 , 1 Now A 3 is mirror image of A 2 in x -axis ⇒ A 3 - 2 ,   - 1 Now finding the intersection point of T 1 = 0 and T 2 = 0 , we get - 3 ,   0 So, Area of quadrilateral A 1 A 2 A 3 A 4 = 1 2 12 + 2 × 5 = 35 square units
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