Question

In Problems 27–38, find an equation for each ellipse. Graph the equation by hand.

Center at (0, 0); focus at (0, 1); vertex at (0, -2)

Short answer

• Equation of ellipse is $\frac{x^2}{3}+\frac{y^2}{4}=1$
  
• Graph of ellipse is

Step-by-step solution

Step 1. Given Information

• Center at (0, 0)
• Focus at (0, 1)
• Vertex at (0, -2)

We want to use the given Information to find the equation of the ellipse with these property. Then graph the ellipse by hand.

Step 2. Find the major axis:

Since the ellipse has its center at the origin (0,0) the major axis must lie either on the y-axis or on the x-axis.

On the major axis lie the foci and vertices of the given ellipse.

Now by noticing that the center, focus and vertex both have the same x-coordinates and that coordinate is zero, and since they have to lie on the major axis, we can conclude that the major axis is equal to the y-axis.

We can also make a drawing to visualize:

Step 3. Finding a and b :

The standard form for an ellipse equation, with its major axis parallel to the x-axis is$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1witha>b$

Therefore, in order to find the equation of the ellipse we need to find a and b .

An ellipse with its major axis being the x-axis has the following vertices:

$(0,-a)and(0,a)$

and the following foci:

$$\begin{gathered} (0,-c)and(0,c) \\ where{c}^2=a^2-b^2 \end{gathered}$$

Since one vertex is given: $(0,-2)$it follows that role="math" localid="1647231559243" $a=2$.

Since the term needed in the equation is a² we finally get

role="math" localid="1647231612231" $a^2=2^2=4$

Similarly, since the focus (0,1) is given, it follows that$c=3$

To find b² use the formula mentioned in the step:

$$\begin{gathered} c^2=a^2-b^2 \\ {b}^2=a^2-c^2 \\ {b}^2=2^2-1^2 \\ {b}^2=4-1 \\ {b}^2=3 \\ b=\pm \sqrt{3} \end{gathered}$$

Step 4. Equation of the ellipse:

Finally, a²=4 and b²=3gives the equation of the ellipse.

From$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

Plug in the values to get$\frac{x^2}{3}+\frac{y^2}{4}=1$

Step 5. Graphing by hand:

To graph the ellipse remember that from the given focus and vertex, we can obtain the other focus and vertex by changing the sign of the y-coordinates.

We can do this because the ellipse is symmetrical with respect to the x-axis.

Therefore the second vertex is $(0,2)$.

Similarity, since the given ellipse is symmetric with respect to the y-axis (its centre in the origin) the x-intercepts are determine by b and they are:

$$\begin{gathered} (-b,0)and(b,0) \\ or \\ (-\sqrt{3},0)and(\sqrt{3},0) \end{gathered}$$

Therefore, by graphing all the vertices and x-intercepts and connecting them we get: