If the line 3x + 4y + k = 0 touches the ellipse 9x^2 + 16y^2 = 144 , then the value of k is.

Step-by-step solution

The equation of the ellipse is 9x^2 + 16y^2 = 144 . Dividing by 144 , we get x^2 16 + y^2 9 = 1 . Here, a^2 = 16 and b^2 = 9 . The equation of the line is 3x + 4y + k = 0 , which can be written as y = - 3 4 x - k 4 . Comparing with y = mx + c , we get m = - 3 4 and c = - k 4 . The condition for the line y = mx + c to touch the ellipse x^2 a^2 + y^2 b^2 = 1 is c^2 = a^2m^2 + b^2 . Substituting the values, we get: (- k 4 )^2 = 16 (- 3 4 )^2 + 9 k^2 16 = 16 ( 9 16 ) + 9 k^2 16 = 9 + 9 = 18 k^2 = 18 16 = 288 k = 12 2 A