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Which one of the following is the common tangent to the ellipses, $\frac{{{x^2}}}{{{a^2} + {b^2}}} + \frac{{{y^2}}}{{{b^2}}}$ $=1\&$ $ \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{a^2} + {b^2}}}$ $=1$
$ay = bx +\sqrt {{a^4} - {a^2}{b^2} + {b^4}} $
$by = ax -\sqrt {{a^4} + {a^2}{b^2} + {b^4}} $
$ay = bx -\sqrt {{a^4} + {a^2}{b^2} + {b^4}} $
$by = ax +\sqrt {{a^4} - {a^2}{b^2} + {b^4}} $
Solution
Equation of a tangent to $\frac{{{x^2}}}{{{a^2} + {b^2}}}\,\, + \,\,\frac{{{y^2}}}{{{b^2}}}\,\, = \,\,1$
$y = mx \pm \,\sqrt {({a^2} + {b^2})\,{m^2}\, + {b^2}} \,$ ….$(1)$
If $(1)$ is also a tangent to the ellipse $\frac{{{x^2}}}{{{a^2}}}\,\, + \,\,\frac{{{y^2}}}{{{a^2} + {b^2}}}\,\, = \,1\,\,$ then
$(a^2 + b^2)m^2 + b^2 = a^2m^2 + a^2 + b^2$ $(using c^2 = a^2m^2 + b^2)$
$b^2m^2 = a^2 \Rightarrow m^2 =\frac{{{a^2}}}{{{b^2}}}\,$ $\Rightarrow m = \pm \frac{a}{b}\,$
$y = \pm \frac{a}{b}\,x\,$ $ \pm \sqrt {({a^2} + {b^2})\frac{{{a^2}}}{{{b^2}}}\, + {b^2}} \,$
$by = \pm ax \pm \sqrt {{a^4} + {a^2}{b^2} + {b^4}} \,$
Note : Although there can be four common tangents but only one of these appears in $B$
