Let the tangent to the ellipse E: x^2 a^2 + y^2 b^2 = 1 ( a > b ) at the point P (3, 16 5 ) intersect the x -axis at the point A ( 25 3 , 0 ) . The distance between the directrices of the ellipse E is equal to

Step-by-step solution

The equation of the tangent to the ellipse x^2 a^2 + y^2 b^2 = 1 at the point P(x₁, y₁) is given by xx₁ a^2 + yy₁ b^2 = 1 . Substituting P (3, 16 5 ) , the equation of the tangent becomes 3x a^2 + 16y 5b^2 = 1 . The x -intercept of this tangent is obtained by putting y = 0 , which gives x = a^2 3 . Given that the tangent meets the x -axis at A ( 25 3 , 0 ) , we have: a^2 3 = 25 3 a^2 = 25 a = 5 Since the point P (3, 16 5 ) lies on the ellipse, it must satisfy the ellipse equation: 3^2 25 + ( 16 5 )^2 b^2 = 1 9 25 +