Let a line L pass through the point of intersection of the lines b x + 10 y - 8 = 0 and 2 x - 3 y = 0 , b ∈ R - 4 3 . If the line L also passes through the point 1 , 1 and touches the circle 17 x 2 + y 2 = 16 , then the eccentricity of the ellipse x 2 5 + y 2 b 2 = 1 is

Step-by-step solution

Line is passing through intersection of b x + 10 y - 8 = 0 and 2 x - 3 y = 0 , So its equation is given by, b x + 10 y - 8 + λ 2 x - 3 y = 0   . . . . . . . . . . 1 And the line is passing through 1 , 1 so λ = b + 2 Now putting the value of λ in equation 1 we get, ⇒ 3 b + 4 x - 3 b - 4 y - 8 = 0 Now line 3 b + 4 x - 3 b - 4 y - 8 = 0 is tangent to circle 17 x 2 + y 2 = 16 So 8 3 b + 4 2 + 3 b - 4 2 = 4 17 ⇒ b 2 = 2 Now eccentricity of the ellipse x 2 5 + y 2 b 2 = 1 will be, e 2 = 1