Question

A common tangent to $9x^2-16y^2=144$ and $x^2+y^2=9$, is

• A.
  
• B.
  
• C.
• D. None of these

Answer 1

The correct option is B


Let $y=mx+c$ be a common tangent to $9x^2-16y^2=144$ and $x^2+y^2=9$
Since, $y=mx+c$ is a tangent to $9x^2-16y^2=144$ Therefore,
$c^2=a^2{m}^2-b^2$ $\Rightarrow$ $c^2=16m^2-9$ …… (i)
Now,$y=mx+c$ is a tangent to $x^2+y^2=9$, therefore $\frac{c}{\sqrt{m^2+1}}=3\Rightarrow c^2=9(1+m^2)$ …… (ii)
From Equations (i) and (ii), we get
$16m^2-9=9+9m^2$ $\Rightarrow$ $m=\pm 3\sqrt{\frac{2}{7}}$ On putting the value of m in Equation (ii), we get $c=\pm \frac{15}{\sqrt{7}}$
Hence, $y=3\sqrt{\frac{2}{7}}x+\frac{15}{\sqrt{7}}$ is a common tangent.