Question

Which of the following is INCORRECT about the ellipse $x^2+4y^2-2x-16y+13=0?$

• A. The latus rectum of the ellipse is $1$
• B. The distance between foci of the ellipse is $2\sqrt{3}$
• C. Eccentricity of the ellipse is $\frac{\sqrt{3}}{2}$
• D. Sum of the focal distances of a point $P(x,y)$ on the ellipse is $4\sqrt{3}$

Answer 1

The correct option is D Sum of the focal distances of a point $P(x,y)$ on the ellipse is $4\sqrt{3}$

$x^2+4y^2-2x-16y+13=0\Rightarrow (x-1{)}^2+4(y-2{)}^2=4\Rightarrow \frac{(x-1{)}^2}{4}+\frac{(y-2{)}^2}{1}=1\Rightarrow a=2,b=1$

Length of latus rectum $=\frac{2b^2}{a}=1$

Eccentricity, $e=\sqrt{1-\frac{b^2}{a^2}}=\frac{\sqrt{3}}{2}$

Distance between focii $=2ae=2\sqrt{3}$

Sum of the focal distance $=2a=4$