Interesting Locus Problem

Geometry Level 4

Consider the ellipse: x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 If the locus of the middle points of a system of parallel chords of this ellipse is a line with slope m m and can be written in the form: y = − b α a β m γ x y = - \frac{b^{\alpha}}{a^{\beta}m^{\gamma}}x Find the value of: α + β + γ \alpha+\beta+\gamma


The answer is 5.

Noor Muhammad Malik by Noor Muhammad Malik , 25, Pakistan

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1 solution

Akshat Sharda
Jan 6, 2017

Let mid point of chord be ( h , k ) (h,k) then the equation of chord is as below.

h x a 2 + k y b 2 = h 2 a 2 + k 2 b 2 \frac{hx}{a^2}+\frac{ky}{b^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}

Given that the chord has slope m m .

∴ m = − b 2 h a 2 k y = − b 2 a 2 m x \therefore m=-\frac{b^2 h}{a^2k} \\ \boxed{y=-\frac{b^2}{a^2 m} x}

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Can anyone tell how the equation came?

Vivek Yadav - 11 months, 3 weeks ago

1 pending report

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