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Write the standard form equation for a hyperbola with center at the origin, vertices at
(0,6) and
(0,-6), and foci at
(0,9) and
(0,-9).

Write the standard form equation for a hyperbola with center at the origin, vertices at $(0,6)$ and $(0,-6)$ , and foci at $(0,9)$ and $(0,-9)$ .

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Write equations of hyperbolas in standard form using properties

Full solution

Q. Write the standard form equation for a hyperbola with center at the origin, vertices at

(0,6)

and

(0,-6)

, and foci at

(0,9)

and

(0,-9)

.

Identify Equation Form: Identify the standard form of the equation for a hyperbola with vertical transverse axis. $\newline$ Standard form: $(y-k)^2/a^2 - (x-h)^2/b^2 = 1$ where $(h,k)$ is the center.
Determine Center: Determine the center $(h,k)$ of the hyperbola. $\newline$ Since the hyperbola is centered at the origin, $h = 0$ and $k = 0$ .
Calculate Semi-Major Axis: Calculate the value of the semi-major axis $a$ . The distance from the center to a vertex is $a$ . The vertices are at $(0,6)$ and $(0,-6)$ , so $a = 6$ .
Find Distance to Focus: Find the distance $c$ from the center to a focus. The foci are at $(0,9)$ and $(0,-9)$ , so $c = 9$ .
Use Relationship to Find $b$ : Use the relationship $c^2 = a^2 + b^2$ to find $b$ . Substitute $a = 6$ and $c = 9$ into the equation. $c^2 = a^2 + b^2$ $9^2 = 6^2 + b^2$ $81 = 36 + b^2$ $b^2 = 81 - 36$ $b^2 = 45$
Write Standard Form Equation: Write the equation of the hyperbola in standard form. $\newline$ Substitute the values of $h$ , $k$ , $a$ , and $b$ into the standard form equation. $\newline$ $(y - 0)^2/6^2 - (x - 0)^2/45 = 1$ $\newline$ $y^2/36 - x^2/45 = 1$

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