A common tangent T to the curves C 1 : x 2 4 + y 2 9 = 1 and C 2 : x 2 42 - y 2 143 = 1 does not pass through the fourth quadrant. If T touches C 1 at x 1 , y 1 and C 2 at x 2 , y 2 , then 2 x 1 + x 2 is equal to _______.

Step-by-step solution

Given curve C 1 : x 2 4 + y 2 9 = 1 and C 2 : x 2 42 - y 2 143 = 1 Now let common tangents to the curves are T 1 : y = m x ± 4 m 2 + 9 and T 2 : y = m x ± 42 m 2 - 143 Since they both are same, So, 4 m 2 + 9 = 42 m 2 - 143 ⇒ 38   m 2 = 152 ⇒ m = ± 2 So, c = ± 4 m 2 + 9 = ± 5 For given tangent not pass through 4 th quadrant So, T : y = 2 x + 5 Now, comparing with point form of tangent x x 1 4 + y y 1 9 = 1 We get, x 1 8 = - 1 5 ⇒ x 1 = - 8 5 And point form of tangent e