Question

The equation of an ellipse, centred at origin and passing through the points $(4,3)$ and $(-1,4),$ is

• A. $7x^2+15y^2=247$
• B. $49x^2+225y^2=105$
• C. $7x^2+15y^2=105$
• D. $7x^2+15y^2=147$

Answer 1

The correct option is A $7x^2+15y^2=247$

Let the equation of the ellipse be
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Since the points $(4,3)$ and $(-1,4)$ lie on the ellipse, we have
$\frac{16}{a^2}+\frac{9}{b^2}=1\dots \left(i\right)$
and $\frac{1}{a^2}+\frac{16}{b^2}=1\dots \left(ii\right)$
From equations $\left(i\right)$ and $\left(ii\right)$, we get
$a^2=\frac{247}{7},b^2=\frac{247}{15}$.

Hence the required equation is
$\frac{x^2}{\left(\frac{247}{7}\right)}+\frac{y^2}{\left(\frac{247}{15}\right)}=1$
$\Rightarrow 7x^2+15y^2=247$