The graph of the equation x2 − xy + y2 = 9 is an ellipse. Find the lines tangent to this curve at the two points where it intersects the x-axis. Show that these lines are parallel.

Answer 1

x^2 - xy + y^2 = 9
2x - x dy/dx - y + 2y dy/dx = 0
dy/dx(2y - x) = y - 2x

dy/dx = (y-2x)/(2y-x)

at x-intercept: y = 0
x^2 - 0 + 0 = 9
x = ± 3

when x = 3, dy/dx = (0-6)/(0-3) = 2
equation of tangent:
y = 2(x-3)
y = 2x - 6

when x = -3 , dy/dx = (0 + 6)/(0+3) = 2
equation at (-3,0)
y = 2(x+3)
y = 2x + 6

since both lines have a slope of 2 , they are parallel