Find an equation for the ellipse whose vertices are at (4,-3) and (4,7), and focus is at (4,4).

Question

asked Jul 20, 2023 199k views

1 Answer

Mightymuke · Answer 1 · 2023-07-24T17:27:46+0000

The vertices of the ellipse are given as

$\begin{gathered} V_2=(4,-3) \\ V_1=(4,7) \end{gathered}$

The focus of the ellipse is given as

The equation of an ellipse is given as

Where the coordinate of the center is

To calculate the coordinate of the center, we will use the formula below

$\begin{gathered} ((x_1+x_2))/(2),((y_1+y_2))/(2) \\ =((4+4))/(2),((-3+7))/(2) \\ =(8)/(2),(4)/(2) \\ (4,2) \\ (h,k)=(4,2) \end{gathered}$

The formula to calculate the value of a is given below

$\begin{gathered} V_1=(h,k+a) \\ V_2=(h,k-a) \end{gathered}$

By comparing coefficients, we will have

$\begin{gathered} k+a=7\ldots\text{.}(1) \\ k-a=-3\ldots\text{.}(2) \end{gathered}$

By adding equations (1) and (2) and solving simultaneously, we will have

$\begin{gathered} 2k=4 \\ \text{divide both sides by 2,} \\ (2k)/(2)=(4)/(2) \\ k=2 \end{gathered}$

By substituting hk=2 in equation 1, we will have

$\begin{gathered} k+a=7 \\ 2+a=7 \\ a=7-2 \\ a=5 \end{gathered}$

The coordinate of the focus,is calculated using the formula below

By substituting the values, we will have

$\begin{gathered} k+c=4 \\ 2+c=4 \\ c=4-2 \\ c=2 \end{gathered}$

The value of will be calculated using the formula below

$\begin{gathered} c^2=a^2-b^2 \\ b^2=a^2-c^2 \end{gathered}$

By substituting the values, we will have

$\begin{gathered} b^2=a^2-c^2 \\ b^2=(5)^2-(2^2 \\ b^2=25-4 \\ b^2=21 \end{gathered}$

By substituting the values of a,b,h and k in the equation of an ellipse, we will have

$\begin{gathered} ((x-h)^2)/(b^2)+((y-k)^2)/(a^2)=1 \\ \frac{(x-4)^2}{21^{}}+\frac{(y-2)^2}{25^{}}=1 \\ \end{gathered}$

Hence,

The equation of the ellipse will be

$\frac{(x-4)^2}{21^{}}+\frac{(y-2)^2}{25^{}}=1$