Question

Find the common tangent to the ellipses $\frac{x^2}{16}+\frac{y^2}{9}=1$ and $\frac{x^2}{9}+\frac{y^2}{16}=1$.

• A. $x+y+5=0$
• B. $x-y+5=0$
• C. $x-y-5=0$
• D. All of the above.

Answer 1

The correct option is D All of the above.

The equation of tangent to a ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is given by $y=mx\pm \sqrt{a^2{m}^2+b^2}$
Therefore, equations of tangent for the two given ellipses are
$y=mx\pm \sqrt{16m^2+9}$
$y=mx\pm \sqrt{9m^2+16}$
As these tangents are the same, we get
$16m^2+9=9m^2+16$
$m^2=1\Rightarrow m=\pm 1$

Therefore, the equation tangent is given by $y=\pm x\pm \sqrt{16+9}$
$\Rightarrow y=\pm x\pm 5$
Hence, all of the above options are correct.