Question

The distance between the foci of the ellipse \[x=3\text{cos}\theta,y=4\text{sin}\theta\] is______.

Options

• \[2\sqrt{7}\]

• \[7\sqrt{2}\]

• \[\sqrt{7}\]

• \[3\sqrt{7}\]

Answer 1

The distance between the foci of the ellipse \[x=3\text{cos}\theta,y=4\text{sin}\theta\] is \[2\sqrt{7}\]

Explanation:

The parametric form \[x=3\cos\theta,y=4\sin\theta\] represents the ellipse:

\[\frac{x^2}{9}+\frac{y^2}{16}=1\]

So \[a^2=16,b^2=9\] (since a > b, the major axis is along the y-axis).

Find c using:

\[c^2=a^2-b^2=16-9=7\Rightarrow c=\sqrt{7}\]

The foci are at \[(0,\pm\sqrt{7})\], so the distance between foci:

\[2c=2\sqrt{7}\]