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Geometry Difficulty 4.8 AIME Prove it

Given the ellipse C: x2a2+y2b2=l\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=l (a>b>0) with its right focus at F(1,0) and eccentricity of 12\frac {1}{2}, 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: x2a2+y2b2=l\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=l (a>b>0) is at F(1,0) and its eccentricity is 12\frac {1}{2}, we have the following system of equations:
{c=1ca=12a2=b2+c2\begin{cases} c=1 \\ \frac {c}{a}= \frac {1}{2} \\ a^{2}=b^{2}+c^{2}\end{cases}
Solving this system, we get a=2 and b=3\sqrt {3}.

Therefore, the equation of the ellipse C is x24+y23=1\frac {x^{2}}{4}+ \frac {y^{2}}{3}=1.

(II) Let M be the point (x₁, y₁) and N be the point (x₂, y₂). From the system of equations:
{y=k(x4)x24+y23=1\begin{cases} y=k(x-4) \\ \frac {x^{2}}{4}+ \frac {y^{2}}{3}=1\end{cases}
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<14\frac {1}{4}.

Hence, x1+x2=32k24k2+3x_{1}+x_{2}= \frac {32k^{2}}{4k^{2}+3} and x₁x₂=64k2124k2+3\frac {64k^{2}-12}{4k^{2}+3}.

Note that when x₁=1 or x₂=1, we get k²=14\frac {1}{4}, which contradicts the problem's condition.

Since k_MF + k_NF = y1x11+y2x21\frac {y_{1}}{x_{1}-1}+ \frac {y_{2}}{x_{2}-1} = k(x14)x11\frac {k(x_{1}-4)}{x_{1}-1} + k(x24)x21\frac {k(x_{2}-4)}{x_{2}-1} = k[2x1x25(x1+x2)+8](x11)(x21)\frac {k[2x_{1}x_{2}-5(x_{1}+x_{2})+8]}{(x_{1}-1)(x_{2}-1)} = k[2(64k2124k2+3)5(32k24k2+3)+8](x11)(x21)\frac {k[2( \frac {64k^{2}-12}{4k^{2}+3})-5( \frac {32k^{2}}{4k^{2}+3})+8]}{(x_{1}-1)(x_{2}-1)} = 0,

we conclude that the angles formed by lines MF and NF with the x-axis are complementary, i.e., the slopes of MF and NF are negative reciprocals of each other\boxed{\text{the slopes of MF and NF are negative reciprocals of each other}}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.