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The tangent at a point P on an ellipse i...

The tangent at a point `P` on an ellipse intersects the major axis at `T ,a n dN` is the foot of the perpendicular from `P` to the same axis. Show that the circle drawn on `N T` as diameter intersects the auxiliary circle orthogonally.

Text Solution

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Let the equation of the ellipse be
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
Let `P(a cos theta, b sin theta)` be a point on the ellipse.
The equation of the tangent at P is `(xcos theta)/(a)+(y sin theta)/(b)=1`
It meets the major axis at `T(a sec theta, 0)`
The coordinates of N are `(a cos theta, 0)`
The equation of the circle with NT as its diameter is `(x-a sec theta)(x-a cos theta)+a^(2)=0`
or `x^(2)y^(2)-ax(sec theta + cos theta)+a^(2)=0`
It cuts the auciliary circle `x^(2)+y^(2)-a^(2)=0` orthogonally as `2((-a sec theta-a cos theta)/(2))(o)+2(0)(0)=a^(2)-a^(2)=0`
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