Ellipse
Geometry
Level
pending
If the locus of the midpoint of the of chords to the ellipse (x/a)^2+(y/b)^2=1 passing through the point (2a,0) is ((x-a)/a)^m+(y/b)^n=1, find m+n
The answer is 4.
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1 solution
Let ( x , y ) be on the ellipse, then
a 2 x 2 + b 2 y 2 = 1
The midpoint of the chord between ( 2 a , 0 ) and ( x , y ) is ( x ′ , y ′ ) = ( a + 2 x , 2 y ) .
Hence, ( x , y ) = ( 2 ( x ′ − a ) , 2 y ′ ) . Substitute this into the equation of the ellipse, you get,
a 2 4 ( x ′ − a ) 2 + b 2 4 y ′ 2 = 1 . Therefore, m = 2 , and n = 2 , making the answer 2 + 2 = 4 .