Question

7.an ellipse with minor axis from (–2, 1) to (–2, 7) and one focus at (2, 4)write it in standard form

Answer 1

`((x+2)^(2))/(25)+((y-4)^(2))/(9)=1`

Explanation

In this case, the minor axis has the coordinates (-2, 1) and (-2, 7), then (referring to the above graph),

h = -2

Given that one focus is placed at (2, 4), then

k = 4

From the point (-2, 7) we can deduce that:

k+b = 7

4+b = 7

b = 7-4

b = 3

From the point (2, 4) we can deduce that:

h+c = 2

-2+c = 2

c = 2+2

c = 4

The relationship between the constants a, b, and c is:

`\begin{gathered} c^2=a^2-b^2 \\ 4^2=a^2-3^2 \\ 16=a^2-9 \\ 16+9=a^2 \\ √(25)=a \\ a=5 \end{gathered}`

Equation of an ellipse

`((x-h)^2)/(a^2)+((y-k)^2)/(b^2)=1`

Substituting with the values previously found:

`\begin{gathered} ((x-(-2))^2)/(5^2)+((y-4)^2)/(3^2)=1 \\ ((x+2)^2)/(25)+((y-4)^2)/(9)=1 \end{gathered}`