Question

The ellipse $x^2+4y^2=4$ is inscribed in a rectangle aligned with the coordinate axes, which is inscribed in another ellipse that passes through the point $(4,0)$. Then the equation of the ellipse is

• A. $x^2+16y^2=16$
• B. $x^2+12y^2=16$
• C. $4x^2+48y^2=48$
• D. $4x^2+64y^2=48$

Answer 1

The correct option is B $x^2+12y^2=16$

Let the equation of the required ellipse,

$\frac{x^2}{16}+\frac{y^2}{b^2}=1$

But the ellipse passes through $(2,1)$

$\Rightarrow \frac{1}{4}+\frac{1}{b^2}=1$

$\Rightarrow \frac{1}{b^2}=\frac{3}{4}$

$b^2=\frac{4}{3}$

Hence the equation is,

$\frac{x^2}{16}+\frac{y^2\times 3}{4}=1$

$\Rightarrow x^2+12y^2=16$