Question

1. (0 points) find the center, vertices, foci, and asymptotes of the hyperbola with equation:

\\9x^2 - 100y^2 - 90x + 800y - 2275 = 0\\

and graph the hyperbola.

center:
vertices:
foci:
asymptotes:

Explanation:

Step1: Group terms

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

$$(9x^2 - 90x) - (100y^2 - 800y) = 2275$$

Step2: Factor coefficients

Factor out the leading coefficients of the squared terms.

$$9(x^2 - 10x) - 100(y^2 - 8y) = 2275$$

Step3: Complete square

Complete the square for both variables.

$$9(x^2 - 10x + 25) - 100(y^2 - 8y + 16) = 2275 + 225 - 1600$$

Step4: Simplify equation

Simplify the right side and write in factored form.

$$9(x - 5)^2 - 100(y - 4)^2 = 900$$

Step5: Standard form

Divide by 900 to get the standard form.

$$\frac{(x - 5)^2}{100} - \frac{(y - 4)^2}{9} = 1$$

Step6: Identify parameters

Identify the center \((h, k)\), \(a^2\), and \(b^2\).

$$(h, k) = (5, 4), \quad a = 10, \quad b = 3$$

Step7: Find vertices

Calculate the vertices using \((h \pm a, k)\).

$$(5 \pm 10, 4) \implies (-5, 4) \text{ and } (15, 4)$$

Step8: Find foci

Calculate \(c = \sqrt{a^2 + b^2}\) and the foci \((h \pm c, k)\).

$$c = \sqrt{100 + 9} = \sqrt{109} \implies (5 \pm \sqrt{109}, 4)$$

Step9: Find asymptotes

Write the equations of the asymptotes.

$$y - k = \pm \frac{b}{a}(x - h) \implies y - 4 = \pm \frac{3}{10}(x - 5)$$

Answer:

Center: \((5, 4)\)
Vertices: \((-5, 4)\) and \((15, 4)\)
Foci: \((5 - \sqrt{109}, 4)\) and \((5 + \sqrt{109}, 4)\)
Asymptotes: \(y - 4 = \pm \frac{3}{10}(x - 5)\)