Chapter 10, Problem 27RE

To determine

To find: The center, vertices, foci, and eccentricity of the ellipse and sketch the ellipse.

Answer to Problem 27RE

The center, vertices, foci, and eccentricity of the ellipse and sketch of the ellipse are $\left(-1,2\right)$ , $\left(-1,2\pm 7\right)$ , $\left(-1,2+2\sqrt{6}\right)$ and $\left(-1,2-2\sqrt{6}\right)$ , $\frac{2\sqrt{6}}{7}$ and shown in figure (1).

Explanation of Solution

Given information:

The equation of the ellipse is

  $\frac{{\left(x+1\right)}^2}{25}+\frac{{\left(y-2\right)}^2}{49}=1$ .

Calculation:

Consider the equation of ellipse.

  $\frac{{\left(x+1\right)}^2}{25}+\frac{{\left(y-2\right)}^2}{49}=1$   ...... (1)

The standard equation of an ellipse with centre $\left(h,k\right)$ and major axis vertical is given by

  $\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1$   ...... (2)

The vertices and foci of this ellipse are at vertices $\left(h,k\pm a\right)$ and foci is $\left(h,k\pm c\right)$ .

Compare equation (1) and (2).

  $\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=\frac{{\left(x+1\right)}^2}{25}+\frac{{\left(y-2\right)}^2}{49}$

From above equation

  $h=-1,k=2,a=7,b=5$

So the center of the ellipse is at the point $\left(-1,2\right)$ and vertices are at $\left(-1,2\pm 7\right)$ , that is at the points $\left(-1,-5\right)$ and $\left(-1,-5\right)$ and $\left(-1,9\right)$ and it is known that in case of ellipse $c$ is related with $a$ and $b$ by the relation.

  $c^2=a^2-b^2$

Substitute the values of $a$ and $b$ in the above equation.

  $\begin{array}{c}{c}^2=49-25\\ c=\pm 2\sqrt{6}\end{array}$

Hence, the foci are at the points $\left(-1,2\pm 2\sqrt{6}\right)$ , that is at the points $\left(-1,2+2\sqrt{6}\right)$ and $\left(-1,2-2\sqrt{6}\right)$ .

The eccentricity of the ellipse is

  $\begin{array}{c}c=ae\\ e=\frac{c}{a}\\ e=\frac{2\sqrt{6}}{7}\end{array}$

Draw the sketch for the ellipse by using the calculated characteristics.

  

Figure-(1)

Therefore, the center, vertices, foci, and eccentricity of the ellipse and sketch of the ellipse are $\left(-1,2\right)$ , $\left(-1,2\pm 7\right)$ , $\left(-1,2+2\sqrt{6}\right)$ and $\left(-1,2-2\sqrt{6}\right)$ , $\frac{2\sqrt{6}}{7}$ and shown in figure (1).