Question

The equation of the ellipse which passes through origin and has its foci at the points $(1,0)$ and $(3,0),$ is

• A. $3x^2+4y^2=x$
• B. $3x^2+y^2=12x$
• C. $x^2+4y^2=12x$
• D. $3x^2+4y^2=12x$

Answer 1

The correct option is D $3x^2+4y^2=12x$

Centre being mid point of the foci is
$(\frac{1+3}{2},0)=(2,0)$
Distance between foci $2ae=2$
$\Rightarrow a^2{e}^2=1$
$\Rightarrow a^2-b^2=1\dots \left(i\right)$
$\because$ foci is on $x-$axis
$\therefore$ the equation of the ellipse be
$\frac{(x-2{)}^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$
As it passes through $(0,0)$
$\therefore \frac{4}{a^2}=1\Rightarrow a^2=4\Rightarrow b^2=3\left[\text{using}\right(i\left)\right]$
Hence equation of required ellipse is
$\frac{(x-2{)}^2}{4}+\frac{y^2}{3}=1$
$\Rightarrow 3x^2+4y^2-12x=0$