Q52 / 150 · MHT CET 2019 (02 May Shift 1) · Mathematics

If $P (x_{1}, y_{1})$ is a point on the hyperbola $x^{2} - y^{2} = a^{2},$ then $S P . S^{'} P =$ ________

A
$\frac{x_{1}^{2} - y_{1}^{2}}{a^{2}}$
B
$\frac{x_{1}^{2} + y_{1}^{2}}{a^{2}}$
C
$x_{1}^{2} - y_{1}^{2}$
Correct answer
D
$x_{1}^{2} + y_{1}^{2}$

Solution

Given Hyperbola

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

is
Rectangular hyperbola, its

e = \sqrt{2}

Then

S (\sqrt{2} a, 0)

and

S' (- \sqrt{2} a, 0)

Also

S^{'} P = (\sqrt{2} x_{1} + a) a n d S P = \sqrt{2} x_{1} - a

Then

(S P) (S^{'} P) = (2 x_{1}^{2} - a^{2}) ∵ x_{1}^{2} - y_{1}^{2} = a^{2}

Hence,

(S P) (S^{'} P) = x_{1}^{2} + y_{1}^{2}

Hence, the correct answer is option: 'D'.

Relevant Questions

The eccentricity of the hyperbola which passes through the points and is25 Apr, 2025 (E)Equation of hyperbola
The area of the region bounded by hyperbola and its latus rectum is02 May, 2024 (M)Equation of hyperbola
The centre of the hyperbola is Ans i.20 Oct, 2020 (M)Equation of hyperbola
The eccentricity of a rectangular hyperbola is19 Oct, 2020 (M)Equation of hyperbola
The eccentricity of the hyperbola is14 Oct, 2020 (M)Equation of hyperbola
The eccentricity of hyperbola 25 x 2 - 9 y 2 = 225 is…2019 (M)Equation of hyperbola