Find the standard form of the equation of the hyperbola with vertices (0±3) and asymptotes (y=±3/2x)
To find the standard form of the equation of a hyperbola given its vertices and asymptotes, we can follow these steps:
Step 1: Identify the center of the hyperbola.
The center of the hyperbola is the midpoint between the vertices. In this case, the vertices are at (0, -3) and (0, 3). The x-coordinate of the center will always be zero because the asymptotes are symmetrical with respect to the y-axis. The y-coordinate of the center is the average of the y-coordinates of the vertices, which is (0 + 3)/2 = 3/2. So, the center of the hyperbola is (0, 3/2).
Step 2: Determine the values of a and b.
The distance between the center and each vertex is called the distance of each vertex from the center and is denoted as 'a'. In this case, a = 3.
Step 3: Determine the equation of the asymptotes.
The equation of the asymptotes is given as y = ±(b/a)x. In this case, the slope of the asymptotes is b/a = (3/2) / 3 = 1/2. Hence, the equation of the asymptotes is y = ±(1/2)x.
Step 4: Determine the value of b.
To find the value of b, we need to determine the distance between the center and a point on either of the asymptotes. By using the distance formula, we can see that the distance from the center (0, 3/2) to the line y = (1/2)x is equal to |(1/2)(0) - (1/2)(3/2) + y - 0| / √(1/2)² + 1².
Simplifying, this becomes |-(3/4) + y| / √(5/4). Since the distance from the center to the asymptote is also equal to b, we can equate these two expressions and solve for b:
|(3/4) - 1/2| / √(5/4) = b
|(3/4) - (2/4)| / √(5/4) = b
|(1/4)| / (√(5)/2) = b
(1/4) * (2/√5) = b
(1/2√5) = b
Step 5: Determine the equation.
Now that we have the center (h, k) = (0, 3/2), a = 3, and b = 1/(2√5), we can write the standard form equation of the hyperbola:
(x - h)² / a² - (y - k)² / b² = 1
Substituting in the values, we get:
x² / 9 - (y - 3/2)² / (1/(2√5))² = 1
Simplifying, we get:
x² / 9 - (y - 3/2)² / (1/20) = 1
Multiplying both sides by 20, we get:
20x² - 10(y - 3/2)² = 180
This is the standard form equation of the hyperbola with the given vertices and asymptotes.