Question

If $\cos \alpha =\frac{2}{3},$ then the range of values of $\varphi$ on the ellipse
$x^2+4y^2=4$ falls inside the circle $x^2+y^2+4x+3=0$ is

• A. $-\alpha ,\alpha$
• B. $0,\alpha$
• C. $\alpha ,\pi$
• D. $\pi -\alpha ,\pi +\alpha$

Answer 1

The correct option is D

$\pi -\alpha ,\pi +\alpha$

The point $\varphi$ on the ellipse $\frac{x^2}{4}+y^2=1$ will be $(2\cos \varphi ,\sin \varphi )$.
It lies inside the circle $x^2+y^2+4x+3=0$ if
$4{\cos }^2\varphi +{\sin }^2\varphi +8\cos \varphi +3<0\Rightarrow 3{\cos }^2\varphi +8\cos \varphi +4<0\Rightarrow (\cos \varphi +2)(3\cos \varphi +2)<0\text{or}-2<\cos \varphi <-\frac{2}{3}\text{or}-1\le \cos \varphi <\cos (\pi -\alpha )(\because cos\varphi \nless -1)\therefore \pi -\alpha <\varphi <\pi +\alpha$ is most appropriate.