Given the ellipse C: (a>b>0) with its right focus at F(1,0) and eccentricity of , and the line 1: y=k(x-4) (k≠0) intersects the ellipse C at two distinct points M and N.
(I) Find the equation of the ellipse C.
(II) Prove that the angles formed by lines MF and NF with the x-axis are complementary.
Solution
(I) Since the right focus of the ellipse C: (a>b>0) is at F(1,0) and its eccentricity is , we have the following system of equations:
Solving this system, we get a=2 and b=.
Therefore, the equation of the ellipse C is .
(II) Let M be the point (x₁, y₁) and N be the point (x₂, y₂). From the system of equations:
we obtain the quadratic equation (4k²+3)x²-32k²x+(64k²-12)=0.
According to the problem, the discriminant Δ=(−32k²)²−4(4k²+3)(64k²−12)>0, which leads to 0<k²<.
Hence, and x₁x₂=.
Note that when x₁=1 or x₂=1, we get k²=, which contradicts the problem's condition.
Since k_MF + k_NF = = + = = = 0,
we conclude that the angles formed by lines MF and NF with the x-axis are complementary, i.e., .
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