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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

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

$ST=SP$

$\Rightarrow Isosceles$

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Riya

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