Question

Family of Ellipses

If $k>o$, the following equation represents an ellipse:

$\frac{x^2}{k}+\frac{y^2}{4+k}=1$

Show that all the ellipses represented by this equation have the same foci, no matter what the value of $k$.

Short answer

Hence proved that for $k>0$, all the ellipses have same foci

Step-by-step solution

Step 1. Given information

Equations of ellipse is given as-

$\frac{x^2}{k}+\frac{y^2}{4+k}=1$

Step 2. Concept used

Each fixed point is called a focus (plural: foci) of the ellipse.

In this problem, we have to show that all the ellipses represented by this equation have the same foci.

Step 3. Calculation

The equation of the ellipse is $\frac{x^2}{k}+\frac{y^2}{4+k}=1$

Since $k>0$ then

$4+k>k$

So the ellipse is vertical since the larger denominator is under the $y^2$term so the foci of the given ellipse are $(0,\pm c)$ where $c=\sqrt{a^2-b^2}$

So in the given ellipse the value of ${\text{a}}^2=4+k\&b^2=k$

So solving for $c$we get

$\begin{array}{l}c=\sqrt{(4+k)-k}\\ c=\sqrt{4}\\ c=2\end{array}$

Therefore the foci are $(0,\pm 2)$for all positive values of $k$.