The equation of an ellipse, whose focus is (1,0) , directrix is x=4 and whose eccentricity is a root of the quadratic equation 2 x^2-3 x+1=0 , is

Step-by-step solution

The given quadratic equation is 2e^2 - 3e + 1 = 0 . Solving for e : 2e^2 - 2e - e + 1 = 0 2e(e - 1) - 1(e - 1) = 0 (2e - 1)(e - 1) = 0 . The roots are e = 1 or e = 1/2 . Since the curve is an ellipse, e Let the focus be S(1, 0) and the directrix be x = 4 . By the definition of a conic section, for any point P(x, y) on the ellipse, SP^2 = e^2 PM^2 , where PM is the perpendicular distance from P to the directrix x - 4 = 0 . (x - 1)^2 + (y - 0)^2 = ( 1 2 )^2 (x - 4)^2 (x - 1)^2 + y^2 = 1 4 (x^2 - 8x + 16) 4(x^2 - 2x +