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Equations of a common tangent to the two hyperbolas $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}$ $= 1 $ $\&$ $\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}}$ $= 1 $ is :
$y = x +\sqrt {{a^2} - {b^2}} $
$y = x -\sqrt {{a^2} - {b^2}} $
$y = - x +\sqrt {{a^2} - {b^2}} $
all of the above
Solution
$\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}}= 1 ….(1)$ and $\frac{{{y^2}}}{{{a^2}}} – \frac{{{x^2}}}{{{b^2}}}= 1….(2)$
Tangent to $(1)$ $ y = mx \pm \sqrt {{a^2}{m^2} – {b^2}} \,$
If this is also tangent to $\frac{{{x^2}}}{{( – {b^2})}}\,\, – \,\,\frac{{{y^2}}}{{( – {a^2})}}\, = \,1$
then $a^2m^2 + b^2 = (-b^2) m^2 – (-a^2) = a^2 -b^2m^2 $
$(a^2 -b^2) m^2 = a^2 -b^2 $
$m = \pm 1$
Hence $4$ common tangents are $y = \pm \,\,x\,\, \pm \,\sqrt {{a^2} – {b^2}} \,$
