Which one of the following is the common tang | vedclass

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Question

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^

Vedclass · Accepted Answer

$by = ax -\sqrt {{a^4} + {a^2}{b^2} + {b^4}} $

Vedclass · Answer

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 $B$

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}}}$ $=1$

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