If equation (10 x-5)²+(10 y-4)²=λ²(3 x+4 y-1)²represents a hyperbola, then

Correct Answer is : λ<−2λ < -2λ<−2 or λ>2λ>2λ>2

Correct Answer is : λ<−2λ < -2λ<−2 or λ>2λ>2λ>2

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BITSAT 2017 Slot 2 Solved Paper

Section: Mathematics

Question : 139 of 150

Marks: +1, -0

If equation

(10 x-5)^{2}+(10 y-4)^{2}=\lambda^{2}(3 x+4 y-1)^{2}

represents a hyperbola, then

Solution:

Given, equation of hyperbola is

(10 x-5)^{2}+(10 y-4)^{2}=\lambda^{2}(3 x+4 y-1)^{2}

can be rewritten as

\frac{\sqrt{\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{2}{5}\right)^{2}}}{\left|\frac{3x+4y-1}{5}\right|}=\left|\frac{\lambda}{2}\right|

This is of the form of

\frac{PS}{PM}=e

Where,

P

is any point on the hyperbola and

S

is a focus and

M

is the point of directrix.

Here,

\left|\frac{\lambda}{2}\right|>1 \Rightarrow |\lambda|>2 (\because e>1)

\Rightarrow \lambda < -2

\lambda>2

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