Question

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

• A. $a y=b x+\sqrt{a^4-a^2 b^2+b^4}$
• B. by $=a x-\sqrt{a^4+a^2 b^2+b^4}$
• C. $a y=b x-\sqrt{a^4+a^2 b^2+b^4}$
• D. by $=a x-\sqrt{a^4-a^2 b^2+b^4}$

Answer 1

Answer: (B)

Let equations of tangent to the two ellipses are
$y=m x \pm \sqrt{\left(a^2+b^2\right) m^2+b^2}$ ......(i)
$y=m x \pm \sqrt{a^2 m^2+a^2+b^2}$......(ii)
On solving (i) and (ii) we get $m=\pm \frac{a}{b}$
Put solve of $m$ in (i) to get the answer.