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The length of a common tangent to the two hyperbolas \mathrm{x^2-3 y^2=3} and \mathrm{y^2-3 x^2=3} is _________ .

Option: 1

5


Option: 2

4


Option: 3

3


Option: 4

2


Answers (1)

The second curve is the image of the first curve in the

line \mathrm{x=y}. The tangent to \mathrm{\frac{x^2}{3}-y^2=1} is

\mathrm{ y=m x \pm \sqrt{3 m^2-1} }                                            ...(i)

The tangent to \mathrm{\frac{y^2}{3}-x^2=1} is

\mathrm{ x=m y \pm \sqrt{3 m^2-1} }                                            ...(ii)

(i), (ii) are identical if \mathrm{\frac{1}{m}=m \, \, or \, \, m= \pm 1}

(i) gives \mathrm{y=x+\sqrt{2}}

which meets the first curve at \mathrm{A\left(\frac{-3}{\sqrt{2}}, \frac{-1}{\sqrt{2}}\right)}

and the second curve at \mathrm{B\left(\frac{1}{\sqrt{2}}, \frac{3}{\sqrt{2}}\right)}

\mathrm{ \therefore A B=\sqrt{(2 \sqrt{2})^2+(2 \sqrt{2})^2}=4 \text {. } }

Posted by

Ramraj Saini

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