If $P(\theta)$ lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ and $S$ and $S^{\prime}$…

If $P(\theta)$ lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ and $S$ and $S^{\prime}$ are foci of the hyperbola, then $S P . S^{\prime} P=$
  1. $a^{2} \tan ^{2} \theta-b^{2} \sec ^{2} \theta$
  2. $a^{2} \tan ^{2} \theta+b^{2} \sec ^{2} \theta$
  3. $a^{2} \sec ^{2} \theta+b^{2} \tan ^{2} \theta$
  4. $a^{2} \sec ^{2} \theta-b^{2} \tan ^{2} \theta$

Solution

$S P \cdot S P^{\prime}=$ By Distance formula we get; $=a^{2} \tan ^{2} \theta+b^{2} \sec ^{2} \theta$

Asked in: MHT CET 2020 (14 Oct Shift 2)

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