A tangent at any point to the ellipse $$4x2+9y2=36$$ is cut by the tangent at the extremities of the major axis at T′ T and '. The circle on TT′ as diameter passes through the point.

Question

A tangent at any point to the ellipse $$4x2+9y2=36$$  is cut by the tangent at the extremities of the major axis at  T′ T  and '. The circle on   TT′ as diameter passes through the point.

Infinity Learn · Accepted Answer

Any point on the ellipse is  $$P(3cos$$⁡θ,$$2sin$$⁡θ$$)Equation$$ of the tangent at P  is $$x3cos$$⁡θ$$+y2sin$$⁡θ$$=1$$ which  meets the tangents  $$x=3$$  and  $$x=$$−$$3$$ at the extremities of the major axis at $$T3$$,$$2(1$$−$$cos$$⁡θ$$)$$$$sin$$⁡θ  and T′−$$3$$,$$2(1+$$$$cos$$⁡θ$$)$$$$sin$$⁡θ Equation of the circle on  TT′  as diameter is     (x$$−$$3)(x+3)+y$$−$$2(1$$−$$cos$$⁡θ$$)$$sin$$⁡θy−$$2(1+$$$$cos$$⁡θ$$)$$$$sin$$⁡θ$$=0$$⇒$$x2+y2$$−$$4sin$$⁡θy−$$5=0$$  which passes through (5$$,$$0)

A tangent at any point to the ellipse
$4 x^{2} + 9 y^{2} = 36$ is cut by the tangent at the extremities of the
major axis at $T^{'}$ $T$ and '. The circle on $T T^{'}$ as diameter passes
through the point.

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A tangent at any point to the ellipse 4x2+9y2=36 is cut by the tangent at the extremities of the major axis at T′ T and '. The circle on TT′ as diameter passes through the point.

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

A tangent at any point to the ellipse
$4 x^{2} + 9 y^{2} = 36$ is cut by the tangent at the extremities of the
major axis at $T^{'}$ $T$ and '. The circle on $T T^{'}$ as diameter passes
through the point.