Question

Analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility.

$4x^2+3y^2+8x-6y=5$

Short answer

The center of ellipse is at $(-1,1)$, vertices are $(-1,3)and(-1,-1)$and the foci are $(-1,2)and(-1,0)$

The required graph is

Step-by-step solution

Step 1. Given Information 

• The given equation is$4x^2+3y^2+8x-6y=5$

Step 2. Calculation

Rewrite the given equation in the general form of equation of ellipse,

$$\begin{gathered} 4x^2+8x+3y^2-6y=5 \\ 4(x^2+2x)+3(y^2-2y)=5 \\ 4(x^2+2x+1)+3(y^2-2y+1)=5+4+3 \\ 4{(x+1)}^2+3{(y-1)}^2=12 \\ \frac{4{(x+1)}^2}{12}+\frac{3{(y-1)}^2}{12}=\frac{12}{12} \\ \frac{(x-{(-1))}^2}{3}+\frac{{(y-1)}^2}{4}=1 \end{gathered}$$

On comparing with the general form of equation of ellipse, we get,

$$\begin{gathered} (h,k)=(-1,1) \\ {a}^2=4,b^2=3 \\ {c}^2=4-3 \\ {c}^2=1 \\ c=1 \end{gathered}$$

The vertices of ellipse are$(h,k\pm a)=(-1,1\pm 2)i.e.,(-1,3)and(-1,-1)$and foci of ellipse are$(h,k\pm c)=(-1,1\pm 1)i.e.,(-1,2)and(-1,0)$

Step 3. Graph

On plotting all the points, we get the required graph of ellipse,