Question

If the coordinates of foci and vertices of an ellipse are (±1, 0) and (±2, 0) respectively, then the minor axis is of length ___________.

Answer 1

For an ellipse,
co-ordinates of foci is (±1, 0)
co-ordinates of vertices is (± 2, 0)
since y-co-ordinate is zero,
∴ equation of ellipse is of form
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1;a>b$
Since vertex in general is given by (± a, 0)
and foci is given by (± ae, 0) ; where e is eccentricity.
∴ a = 2 and ae = 1
$$\begin{gathered} \Rightarrow e=\frac{1}{2} \\ \Rightarrow e^2=1-\frac{b^2}{a^2} \\ \mathrm{i}.\mathrm{e}{\left(\frac{1}{2}\right)}^2=1-\frac{b^2}{4} \\ \mathrm{i}.\mathrm{e}\frac{b^2}{y}=1-\frac{1}{4}=\frac{3}{4} \\ \mathrm{i}.\mathrm{e}{b}^2=3 \\ \mathrm{i}.\mathrm{e}b=\sqrt{3} \\ \therefore \mathrm{length}\mathrm{of}\mathrm{minor}\mathrm{axis}\mathrm{is}2\sqrt{3} \end{gathered}$$