Question

For the ellipse 4x2+y2−8x+2y+1=04x2+y2−8x+2y+1=0 which of the following statements are correct:

• A. Foci are $(-1,-1\pm \sqrt{3}),$ Directrices are $y=-1\pm \frac{4}{\sqrt{3}}$
• B. Foci are $(1,1\pm \sqrt{3}),$ Directrices are $y=1\pm \frac{4}{\sqrt{3}}$
• C. Foci are $(1,-1\pm \sqrt{3}),$ Directrices are $y=-1\pm \frac{4}{\sqrt{3}}$
• D. Foci are $(1,-1\pm \sqrt{3}),$ Directrices are $y=1\pm \frac{4}{\sqrt{3}}$

Answer 1

The correct option is C Foci are $(1,-1\pm \sqrt{3}),$ Directrices are $y=-1\pm \frac{4}{\sqrt{3}}$

$4x^2+y^2-8x+2y+1=0$
$4(x^2-2x)+(y^2+2y)=-1$
$4(x^2-2x+1)+(y^2+2y+1)=-1+5=4$
$4(x-1{)}^2+(y+1{)}^2=4$
$\frac{{(x-1)}^2}{1}+\frac{{(y+1)}^2}{4}=1$
$\Rightarrow a^2=1,b^2=4$ Clearly here $a^2<b^2$ so the axis of the ellipse is parallel to y-axis
Now, $e=\sqrt{1-\frac{a^2}{b^2}}=\frac{\sqrt{3}}{2}\therefore$ Foci $(1,-1\pm \sqrt{3}),$ Directrices $y=-1\pm \frac{4}{\sqrt{3}}.$