A chord of the ellipse 2x^2 + y^2 = 18 is parallel to the line y = 2x , and its midpoint lies on the line x - y = 2 . The square of the length of this chord is

Step-by-step solution

Let the midpoint of the chord be (h, k) . The equation of the chord is given by T = S₁ : 2hx + ky = 2h^2 + k^2 The slope of this chord is - 2h k . Since it is parallel to y = 2x , we have: - 2h k = 2 k = -h The midpoint (h, k) lies on the line x - y = 2 : h - k = 2 Substituting k = -h , we get: h - (-h) = 2 2h = 2 h = 1 Thus, k = -1 . The midpoint is (1, -1) . The equation of the chord becomes: 2(1)x + (-1)y = 2(1)^2 + (-1)^2 2x - y = 3 y = 2x - 3 To find the points of intersection, substitute y = 2x - 3 into the e