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Show that the double ordinate of the auxilliary circle of an ellipse passing through the focus is equal to the minor axis of the ellipse.

Show that the double ordinate of the auxilliary circle of an ellipse passing through the focus is equal to the minor axis of the ellipse.

Grade:11

1 Answers

Rajat
213 Points
2 years ago
Let the focus of the ellipse be F, the major axis be 'a' and minor axis be 'b'. Let PFP' be the double ordinate of the auxiliary circle where P and P' are points on the circle. let O be the origin then, OF = ae= sqrt(a^2-b^2).
triangle OFP is right angled. Thus OP^2-OF^2=PF^2
or, PF^2 = a^2-(a^2-b^2)= b^2
so, PF = b, therefore PFP' = 2b = minor axis

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