If P ( x _ 1 , y _ 1 ) is a point on the hyperbola x^ 2 - y^ 2 = a^ 2 , then SP . S^ ' P = ________

Correct answer: D. x _ 1 ^ 2 + y _ 1 ^ 2 Given Hyperbola x^ 2 - y^ 2 = a^ 2 is Rectangular hyperbola, its e = 2 Then S ( 2 a, 0) and S' (- 2 a, 0) Also S^ ' P = ( 2 x _ 1 + a)andSP = 2 x _ 1 - a Then (SP)(S^ ' P) = (2 x _ 1 ^ 2 - a^ 2 ) ∵ x _ 1 ^ 2 - y _ 1 ^ 2 = a^ 2 Hence, (SP)( S^ ' P) = x _ 1 ^ 2 + y _ 1 ^ 2

JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Hyperbola

JEE Main · Mathematics · Hyperbola

If P ( x _ 1 , y _ 1 ) is a point on the hyperbola x^ 2 - y^ 2 = a^...

P \left({x}_{1}, {y}_{1}\right) is a point on the hyperbola

x^{2} - y^{2} = a^{2},

then

SP . S^{'}P = ________

Options

A $\dfrac{{x}_{1}^{2} - {y}_{1}^{2}}{a^{2}}$
B $\dfrac{{x}_{1}^{2} + {y}_{1}^{2}}{a^{2}}$
C ${x}_{1}^{2} - {y}_{1}^{2}$
D ${x}_{1}^{2} + {y}_{1}^{2}$

Correct answer

D. ${x}_{1}^{2} + {y}_{1}^{2}$

Step-by-step solution

Given Hyperbola

x^{2} - y^{2} = a^{2}

is
Rectangular hyperbola, its

e = \sqrt{2}

Then

S \left(\sqrt{2}a, 0\right)

and

S' \left(-

\sqrt{2}

a, 0\right)

Also

S^{'}P = \left(\sqrt{2}{x}_{1} +

a\right)andSP =

\sqrt{2}{x}_{1} -

<br/>Then \left(SP\right)\left(S^{'}P\right) = \left(2{x}_{1}^{2} - a^{2}\right) ∵ {x}_{1}^{2} - {y}_{1}^{2} = a^{2}

Hence, \left(SP\right)\left(

S^{'}P\right) = {x}_{1}^{2} + {y}_{1}^{2}

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