Question

The number of points at which the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and the circle $x^2+y^2-8x+15=0$ intersect is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B $1$

$\frac{x^2}{25}+\frac{y^2}{9}=1$
Point on the ellipse,
$=(5\cos \theta ,3\sin \theta )$
Putting this on the circle,
$25{\cos }^2\theta +9{\sin }^2\theta -40\cos \theta +15=0\Rightarrow 16{\cos }^2\theta -40\cos \theta +24=0\Rightarrow 2{\cos }^2\theta -5\cos \theta +3=0\Rightarrow (2\cos \theta -3)(\cos \theta -1)=0\Rightarrow \cos \theta =1\Rightarrow \theta =2n\pi$

The point will be $(5,0)$
Hence they will touch at one point only.