Chapter 9.6, Problem 14OE

To determine

To calculate: To find the graph the equation

Explanation of Solution

Given information: Equation is $\frac{{\left(x+1\right)}^2}{4}+\frac{{\left(y-2\right)}^2}{1}=1$

Formula Used:

Standard form of the equation of ellipse with center $\left(h,k\right)$ and major axis parallel to x-axis is:

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

Where, $a=$ x-intercept, $b=$ y-intercept, $c^2=a^2-b^2$

When, $a>b$ , foci lies on x-axis and coordinates of foci are $\left(h\pm c,k\right)$

When $b>a$ , foci lies on y-axis and coordinates of foci are $\left(h,k\pm c\right)$

Calculation:

Equation is given as follows:

  $\frac{{\left(x+1\right)}^2}{4}+\frac{{\left(y-2\right)}^2}{1}=1$

Rewriting the equation as follows:

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

Comparing the equation with standard form of the equation of ellipse $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$ :

  $\begin{array}{l}a=1\\ b=1\\ \left(h,k\right)=\left(-1,2\right)\end{array}$

Since $a>b$ , thus foci lies on x-axis.

Plotting the graph of ellipse:

  

Conclusion:

Hence, matching graph is