Question

write the equation of the ellipse (4x^2 + 9y^2 + 8x - 36y + 4 = 0) in standard form

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

where:
(h =)
(k =)
(a =)
(b =)

Explanation:

Step1: Group terms

Group the \(x\) and \(y\) terms together.

$$(4x^2 + 8x) + (9y^2 - 36y) = -4$$

Step2: Factor coefficients

Factor out the leading coefficients from each group.

$$4(x^2 + 2x) + 9(y^2 - 4y) = -4$$

Step3: Complete squares

Add the necessary constants inside the parentheses to complete the squares.

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

Step4: Simplify equation

Simplify both sides of the equation.

$$4(x + 1)^2 + 9(y - 2)^2 = 36$$

Step5: Divide by 36

Divide both sides by 36 to get the standard form.

$$\frac{(x + 1)^2}{9} + \frac{(y - 2)^2}{4} = 1$$

Step6: Identify parameters

Compare with the standard form \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\).

$$h = -1, \quad k = 2, \quad a = 3, \quad b = 2$$

Answer:

\(h = -1\)
\(k = 2\)
\(a = 3\)
\(b = 2\)