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?

Step-by-step solution

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 &