Which one of the following is the common tang | vedclass

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(B) The equation of a tangent to the ellipse $\frac{x^2}{a^2 + b^2} + \frac{y^2}{b^2} = 1$ is given by $y = mx \pm \sqrt{(a^2 + b^2)m^2 + b^2}$.If thi

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

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(B) The equation of a tangent to the ellipse $\frac{x^2}{a^2 + b^2} + \frac{y^2}{b^2} = 1$ is given by $y = mx \pm \sqrt{(a^2 + b^2)m^2 + b^2}$.If this line is also a tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{a^2 + b^2} = 1$,then its condition of tangency $c^2 = A^2m^2 + B^2$ must be satisfied,where $A^2 = a^2$ and $B^2 = a^2 + b^2$.Thus,$(a^2 + b^2)m^2 + b^2 = a^2m^2 + (a^2 + b^2)$.Simplifying this,we get $a^2m^2 + b^2m^2 + b^2 = a^2m^2 + a^2 + b^2$,which implies $b^2m^2 = a^2$.Therefore,$m^2 = \frac{a^2}{b^2}$,so $m = \pm \frac{a}{b}$.Substituting $m = \pm \frac{a}{b}$ into the tangent equation:$y = \pm \frac{a}{b}x \pm \sqrt{(a^2 + b^2)\frac{a^2}{b^2} + b^2} = \pm \frac{a}{b}x \pm \sqrt{\frac{a^4 + a^2b^2 + b^4}{b^2}}$.Multiplying by $b$,we get $by = \pm ax \pm \sqrt{a^4 + a^2b^2 + b^4}$.Comparing this with the given options,$by = ax - \sqrt{a^4 + a^2b^2 + b^4}$ is a common tangent.

Which one of the following is the common tangent to the ellipses $\frac{x^2}{a^2 + b^2} + \frac{y^2}{b^2} = 1$ and $\frac{x^2}{a^2} + \frac{y^2}{a^2 + b^2} = 1$?

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