Ellipse
Geometry
Level
pending
If tan(A)*tan(B)=-(a/b)^2, then the chord joining the two points whose eccentric angles are A and B on the ellipse (x/a)^2+(y/b)^2=1 will subtend a right angle at
end of major axis
centre
end of minor axis
focus
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1 solution
The point on the ellipse corresponding to eccentric angle A is:
P = ( a cos A , b sin A )
And , the point on the ellipse corresponding to eccentric angle B is:
Q = ( a cos B , b cos B )
The center of the ellipse is point O = ( 0 , 0 )
The angle subtended by PQ at the origin will be right if O P ⋅ O Q = 0
O P ⋅ O Q = a 2 cos A cos B + b 2 sin A sin B = a 2 cos A cos B ( 1 + ( b 2 / a 2 ) tan A tan B ) = 0
Hence, the chord PQ subtends a right angle at the center of the ellipse.