Equations of a common tangent to the two hype | vedclass

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Question

$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}= 1 ....(1)$ and $\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}}= 1....(2)$
Tangent to $(1)$ $ y

Vedclass · Accepted Answer

all of the above

Vedclass · Answer

$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}= 1 ....(1)$ 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 $y =  \pm \,\,x\,\, \pm \,\sqrt {{a^2} - {b^2}} \,$

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 :

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