Question

Write the equation of the ellipse in standard form. Show all work.4x^2+9y^2-32x+36y+64=0

Answer 1

To convert the given general form of the equation of the ellipse to standard form, here are the steps:

1. Group the terms with the same variable. Transfer the constant term on the other side of the equation.

`(4x^2-32x)+(9y^2+36y)=-64`

2. Extract a common factor on each group.

`4(x^2-8x)+9(y^2+4y)=-64`

3. Apply completing the square method in each group in parenthesis to make it a perfect square trinomial. Divide the middle term by 2 and square it. The result will be the third term in each group.

`4(x^2-8x+16)+9(y^2+4y+4)=-64`

4. Multiply the constant term in each group to each respective factor and add the results to the other side of the equation.

`\begin{gathered} 4*16=64 \\ 9*4=36 \\ 4(x^2-8x+16)+9(y^2+4y+4)=-64+64+36 \\ 4(x^2-8x+16)+9(y^2+4y+4)=36 \end{gathered}`

5. Rewrite each group of trinomial to binomial using its factors.

`4(x-4)^2+9(y+2)^2=36`

6. Lastly, divide both sides of the equation by the constant term 36.

`\begin{gathered} (4(x-4)^2)/(36)+(9(y+2)^2)/(36)=(36)/(36) \\ ((x-4)^2)/(9)+((y+2)^2)/(4)=1 \end{gathered}`

Hence, the equation of the ellipse in standard form is:

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