First slide

Hyperbola in conic sections

Question

$P$ is a point on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 S$ $S and S^{'}$ are its foci.
Statement-1: Product of the lengths of the perpendiculars from S and $S^{'}$ on the tangent at P is equal to
Statement-2: $|P S - P S^{'}| = 2 a$ .

Moderate

A

STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is NOT a correct explanation for STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

Let $P (a \sec θ, b \tan θ)$ equation of the tangent at P is $\frac{x}{a} \sec θ - \frac{y}{b} \tan θ = 1$
Product of the lengths of the perpendiculars from
$\begin{array}{l} |\frac{(e \sec θ - 1) (e \sec θ + 1)}{\frac{\sec^{2} θ}{a^{2}} + \frac{\tan^{2} θ}{b^{2}}}| \\ = \frac{a^{2} b^{2} (e^{2} \sec^{2} θ - 1)}{a^{2} [(e^{2} - 1) \sec^{2} θ + \tan^{2} θ]} = b^{2} \end{array}$
So statement-1 is trure.
For statement-2, by definition of hyperbola distance of P from a focus is e times its distance from the corresponding directrix.
$\begin{array}{l} So P S = e (x - \frac{a}{e}) and P S^{'} = e (x + \frac{a}{e}) \\ \Rightarrow |P S - P S^{'}| = 2 a \end{array}$
Thus statement-2 is also true but does not justify statement-1.

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