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Find the equation of the hyperbola with the vertices
(0,3)(10,3) and the asymptotes
y=2//5x+1 and
y=-2//5x+5

$6$ . Find the equation of the hyperbola with the vertices $(0,3)(10,3)$ and the asymptotes $y=2 / 5 x+1$ and $y=-2 / 5 x+5$

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Write equations of hyperbolas in standard form using properties

Full solution

Q.

6

. Find the equation of the hyperbola with the vertices

(0,3)(10,3)

and the asymptotes

y=2 / 5 x+1

and

y=-2 / 5 x+5

Identify Hyperbola Equation Form: Identify the standard form of the equation for a hyperbola with a horizontal transverse axis. $\newline$ Standard form of equation for a hyperbola with a horizontal transverse axis: $\newline$ $(x-h)^2/a^2 - (y-k)^2/b^2 = 1$
Determine Center of Hyperbola: Determine the center $(h, k)$ of the hyperbola. $\newline$ The vertices are $(0, 3)$ and $(10, 3)$ . The center is the midpoint of the line segment joining the vertices. $\newline$ $h = (0 + 10)/2 = 5$ $\newline$ $k = (3 + 3)/2 = 3$
Calculate Semi-Major Axis: Calculate the value of the semi-major axis $a$ . $a$ is the distance from the center to a vertex. $a = \frac{10}{2} = 5$
Determine Asymptote Slope: Determine the slope of the asymptotes to find the value of $b$ . The slopes of the asymptotes for a horizontal hyperbola are given by $\pm\frac{b}{a}$ . The given slopes are $\pm\frac{2}{5}$ . $\frac{b}{a} = \frac{2}{5}$ Since we already know $a = 5$ , we can solve for $b$ . $b = \left(\frac{2}{5}\right) \cdot a$ $b = \left(\frac{2}{5}\right) \cdot 5$ $b = 2$
Write Standard Form Equation: Write the equation of the hyperbola in standard form using the values of $h$ , $k$ , $a$ , and $b$ . Substitute the values into $(x-h)^2/a^2 - (y-k)^2/b^2 = 1$ . $(x - 5)^2/5^2 - (y - 3)^2/2^2 = 1$ Simplify the equation. $(x - 5)^2/25 - (y - 3)^2/4 = 1$

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