Question

The number of tangents to the circle $x^2+y^2=3$ that are normals to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}$ is

• A. one
• B. two
• C. three
• D. zero

Answer 1

The correct option is A one

Let y = mx+c is tangent to $x^2+y^2=3$

Then by condition of tangeney

$\left|\frac{c}{\sqrt{m^2+1}}\right|=\sqrt{3}$

$\Rightarrow c^2=3(m^2+1)$ __(1)

$y=mx+c$ is Normal to $\frac{x^2}{9}+\frac{y^2}{4}=1$

If

$c=\frac{(b^2-a^2)m}{\sqrt{a^2+b^2{m}^2}}$

$\Rightarrow c=\frac{(4-9)m}{\sqrt{9+4m2}}$

$\Rightarrow c^2=\frac{25m^2}{4m^2+9}$

$\Rightarrow 3(m^2+1)(4m^2+9)=25m^2$

let $m^2=t$

$\Rightarrow 3(t+1)(4t+9)=25t$

since, $D<0.$

Hence, There is no root :

Hence, There is no tangent to circle which is

Normal to ellipse.