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  • The asymptotes to the hyperbola  are   <div class='qna-optio

The asymptotes to the hyperbola x y=h x+h y are 

 

Option: 1

x=k, y=h


Option: 2

x=h, y=h


Option: 3

x=h, y=h


Option: 4

x=k, y=k


Answers (1)

best_answer

(a) The equation of asymptotes is 

                            x y-h x-k y+\lambda=0\: \: \: \: \: \: \: \: \: \: \: \: ....(i)

where, \lambda is a constant and Eq. (i) represents a pair of straight line.

Here,           A=0, B=0, C=\lambda, 2 H=1,2 G=-h \text { and } 2 F=-k

Then,          A B C+2 F G H-A F^2-B G^2-C H^2=0

\begin{aligned} & \Rightarrow 0+2\left(-\frac{k}{2}\right)\left(-\frac{h}{2}\right)\left(\frac{1}{2}\right)-0-0-\lambda \cdot \frac{1}{4}=0 \\ \\& \Rightarrow \quad \frac{h k}{4}=\frac{\lambda}{4} \Rightarrow \lambda=h k \end{aligned}

On putting \lambda=hk in Eq. (i), we get

\begin{aligned} \Rightarrow & & x y-h x-k y+h k & =0 \\ \\\Rightarrow & & (x-k)(y-h) & =0 \end{aligned}

So, the asymptotes are x=k and y=h

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Ritika Jonwal

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