Question

write the equation of the hyperbola (16x^2 - 36y^2 - 64x - 216y - 836 = 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.

$$(16x^2 - 64x) - (36y^2 + 216y) = 836$$

Step2: Factor coefficients

Factor out the leading coefficients from each group.

$$16(x^2 - 4x) - 36(y^2 + 6y) = 836$$

Step3: Complete squares

Add the necessary constants inside the parentheses.

$$16(x^2 - 4x + 4) - 36(y^2 + 6y + 9) = 836 + 16(4) - 36(9)$$

Step4: Simplify equation

Simplify both sides of the equation.

$$16(x - 2)^2 - 36(y + 3)^2 = 576$$

Step5: Divide constants

Divide both sides by 576 to set to 1.

$$\frac{(x - 2)^2}{36} - \frac{(y + 3)^2}{16} = 1$$

Step6: Identify parameters

Compare with standard form to find \(h\), \(k\), \(a\), and \(b\).

$$h = 2, \quad k = -3, \quad a = 6, \quad b = 4$$

Answer:

h = 2
k = -3
a = 6
b = 4