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Q   is a point on the auxiliary circle corresponding to the point P  of the ellipse  \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 . If T is the foot of the perpendicular dropped from the focus S onto the tangent to the auxiliary circle at Q then the \Delta SPT  is:

Option: 1

Isosceles


Option: 2

Equilateral


Option: 3

Right angled


Option: 4

Right isosceles


Answers (1)

best_answer

 

Equation of tangent in parametric form -

x\cos \alpha +y\sin \alpha = a

 

- wherein

Tangent a point \left ( a\cos \alpha ,a\sin \alpha \right ) to x^{2}+y^{2}=a^{2}

 

 

Standard equation -

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}= 1
 

- wherein

a\rightarrow Semi major axis

b\rightarrow Semi minor axis

 

 

Tangent at Q is , x \cos\theta+y\sin\theta=a

ST=\left | a\:e\:\cos\theta-a \right |=a(1-e\cos\theta)

also    

SP=ePM=e\left ( \frac{a}{e}-a\cos\theta \right )=a(1-e\cos\theta)

ST=SP

\Rightarrow Isosceles

 

Posted by

Riya

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