Chapter 11, Problem 16RE

To determine

To find: The center, vertices, foci and length of major and minor axis of ellipse and represent these through a graph.

Answer to Problem 16RE

Center: $\left(-1,2\right)$

Foci: $=\left(-1,0\right)\text{ and }\left(-1,4\right)$

Length of major axis $=4\sqrt{2}$

Length of minor axis $=4$

Vertices $=\left(-1,0\right)\text{ and }\left(-1,4\right)$

Explanation of Solution

Given information:

Given equation of the ellipse

  $2x^2+y^2=2+4\left(x-y\right)$

Calculation:

If the equation of the ellipse:-

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

Then center of the ellipse is $\left(h,k\right)$

If $a>b$

Length of the major axis $=2a$

Length of the minor axis $=2b$

Coordinate of the vertices $\Rightarrow \left(h\pm a,k\right)$

Coordinates of the foci $\Rightarrow \left(h\pm c,k\right)$

  $c^2=a^2-b^2$

If $b>a$

Length of the major axis $=2b$

Length of the minor axis $=2a$

Coordinate of the vertices $\Rightarrow \left(h,k\pm a\right)$

Coordinate of the foci $\Rightarrow \left(h,k\pm c\right)$

  $c^2=a^2+b^2$

Convert $2x^2+y^2=2+4\left(x-y\right)$ to

  $\begin{array}{l}\frac{{\left(x-a\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1\\ 2x^2+y^2=2+4x-4y\\ 2x^2+y^2-4x+4y=2\\ {\left(\sqrt{2}x+\sqrt{2}\right)}^2+{\left(y-2\right)}^2=8\\ \frac{{\left(x+1\right)}^2}{4}+\frac{{\left(y-2\right)}^2}{8}=-1\\ \text{ Here }a=2,b=2\sqrt{2}\\ h=-1,k=2\end{array}$

  $b>a$

Center of the ellipse $\left(h,k\right)=\left(-1,2\right)$

  $\therefore$ Then length of the major axis $=2b=2\left(2\sqrt{2}\right)=4\sqrt{2}$

Length of the minor axis $=2a=2\left(2\right)=4$

Coordinate of vertices $=\left(h,k\pm a\right)$

  $=\left(-1,2\pm 2\right)$

  $\Rightarrow \left(-1,0\right)\text{ and }\left(-1,4\right)$

Coordinate of foci $\Rightarrow \left(h,k\pm c\right)$

  $\begin{array}{l}{c}^2=b^2-a^2\\ c^2=8-4\Rightarrow c=2\\ =\left(-1,2\pm 2\right)\\ =\left(-1,0\right)\text{ and }\left(-1,4\right)\end{array}$

  

Interpretation :

Through the equation of the ellipse $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$ , the center, vertices, foci and length of major and minor axis of ellipse have been determined.