Question

Writing an equation of an ellipse given the foci and major axis length

Answer 1

Given

Major axis of length = 12

Foci at ( 9 ,1 ) and ( -1, 1 )

Find

Equation of an ellipse

Step-by-step explanation

As we know , major axis = 2a = 12

thus a = 6

the midpoint between the foci is the center , so

`\begin{gathered} C:((8)/(2),(2)/(2)) \\ C:(4,2) \end{gathered}`

the distance between the foci is equal to 2c

`\begin{gathered} 2c=√(\left(9+1\right)^2+0^2) \\ 2c=10 \\ c=5 \end{gathered}`

now,

`\begin{gathered} c^2=a^2-b^2 \\ b^2=a^2-c^2 \\ b^2=36-25 \\ b^2=11 \end{gathered}`

so , the equation of an ellipse is

`\begin{gathered} ((x-h)^2)/(b^2)+((y-k)^2)/(a^2) \\ ((x-4)^2)/(11)+((y-2)^2)/(36) \end{gathered}`

Final Answer

The equation of an ellipse

`((x-4)^(2))/(11)+((y-2)^(2))/(36)`