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The asymptotes of the hyperbola  \mathrm{x y=h x+k y}  are

Option: 1

\mathrm{x-k=0 \text { and } y-h=0}


Option: 2

\mathrm{x+h=0 \text { and } y+k=0}


Option: 3

\mathrm{x-k=0 \text { and } y+h=0}


Option: 4

\mathrm{x+k=0 \text { and } y-h=0}


Answers (1)

best_answer

The given hyperbola is

\mathrm{x y-h x-k y=0}                                ............(1)

The equation of asymptotes is given by

\mathrm{x y-h x-k y+c=0}                      .............(2)

Equation (2) gives a pair of straight lines. So,

        \mathrm{\left|\begin{array}{lll} A & H & G \\ H & B & F \\ G & F & C \end{array}\right|=0}

or    \mathrm{\left|\begin{array}{ccc} 0 & 1 / 2 & -h / 2 \\ 1 / 2 & 0 & -k / 2 \\ -h / 2 & -k / 2 & c \end{array}\right|=0}

        \mathrm{\begin{aligned} & \frac{h k}{8}+\frac{h k}{8}-\frac{c}{4}=0 \\ & c=h k \end{aligned}}

Hence, the asymptotes are

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

 

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