Question

Use the image of the ellipse below to create the corresponding equation in standard form and then answer the following questions. If a value is a non-integer type your answer as a reduced fraction.ellipse center (1,3), major axis parallel to X axis vertices at (-2,3) and (4,3). Minor axis parallel to Y axis vertices at (1,2) and (1,4)The standard form for an ellipse whose major axis is parallel to the x axis is:\frac{(x-h)^2}{a^2} +\frac{(y-k)^2}{b^2}=1 Where a>b and the point (h,k) is the center of the ellipse.The value for h is: AnswerThe value for k is: AnswerThe value for a is: AnswerThe value for b is: Answer

Answer 1

Solution:

Given the graph of the ellipse as shown below:

The standard form of an ellipse whose major axis is parallel to thex-axis is

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

The length of the major axis equals 2a.

Thus,

$\begin{gathered} 2a=4-(-2) \\ \Rightarrow2a=6 \\ \text{divide both sides by 2} \\ \text{thus,} \\ a=\frac{6}{2} \\ \Rightarrow a=3 \end{gathered}$

We can determine the center (h,k) using the midpoint formula expressed as

$\begin{gathered} (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \\ \Rightarrow(h,k)=(\frac{-2+4}{2},\frac{3+3}{2}) \\ \Rightarrow(h,k)=(1,3) \end{gathered}$

The co-vertices of the graphed ellipse is

$(h,k\pm b)$

This implies that

$\begin{gathered} (h,\text{ k+b)}\Rightarrow(1,4) \\ (h,\text{ k-b)}\Rightarrow2 \end{gathered}$

Thus, when

$\begin{gathered} k+b=4 \\ \text{where k=}3 \\ we\text{ have} \\ 3+b=4 \\ \Rightarrow b=4-3 \\ \therefore b=1 \end{gathered}$

Hence,

$\begin{gathered} \text{the value for h is 1} \\ \text{the value }for\text{ k is }3 \\ \text{the value for }a\text{ is 3} \\ \text{the value for }b\text{ is 1} \end{gathered}$