A chord of the ellipse x^2 16 + y^2 4 = 1 has its mid-point on the line y = x . If this chord is also tangent to the circle x^2 + y^2 = 1 , then the length of the chord is:

Step-by-step solution

Let the mid-point of the chord be P(h, h) since it lies on the line y = x . The equation of the chord with mid-point (h, h) is given by T = S₁ : hx 16 + hy 4 = h^2 16 + h^2 4 x + 4y = 5h This chord is tangent to the circle x^2 + y^2 = 1 . The perpendicular distance from the origin (0,0) to the line must be equal to the radius of the circle, which is 1 . |-5h| 1^2 + 4^2 = 1 25h^2 = 17 h^2 = 17 25 Now, to find the length of the chord, we find the points of intersection of the line x = 5h - 4y and the ellipse x^2 + 4y