what are the slopes of the asymptotes of the hyperbola: (x^2)/9 - (y^2)/4 = 1
a=3 and b=2
asymptote equation in standard form are
y = ± bx/a
so y = ± (2/3)x
To find the slopes of the asymptotes of the hyperbola, we can rewrite the equation in the standard form of a hyperbola, which is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1,
where (h, k) represents the center of the hyperbola, and a and b are the distances from the center to the vertices. In this case, we have:
(x^2) / 9 - (y^2) / 4 = 1.
Comparing this to the standard form, we can see that:
h = 0 (since there is no (x - h) term)
k = 0 (since there is no (y - k) term)
a^2 = 9 (taking the square root, a = 3)
b^2 = 4 (taking the square root, b = 2).
The slopes of the asymptotes can be found using the formula:
m = ±(b / a).
Plugging in the values of a and b, we have:
m = ±(2 / 3),
So the slopes of the asymptotes of the hyperbola are +(2 / 3) and -(2 / 3).
To find the slopes of the asymptotes of a hyperbola, it is necessary to examine its equation and compare it to the standard form of a hyperbola equation.
The standard form of a hyperbola equation is given by:
((x – h)^2)/a^2 - ((y – k)^2)/b^2 = 1,
Where (h, k) represents the center of the hyperbola. In this case, the given hyperbola equation is:
(x^2)/9 - (y^2)/4 = 1.
By comparing this equation to the standard form, we can determine that the center of the hyperbola is at the origin (0, 0), since there are no values outside the parentheses. Additionally, the value of a^2 is 9, which means that the distance from the center to the vertices along the x-axis is the square root of 9, which is 3. Similarly, the value of b^2 is 4, meaning that the distance from the center to the vertices along the y-axis is the square root of 4, which is 2.
The slope of the asymptotes of a hyperbola is given by the formula:
m = ± b/a,
Where b represents the semi-minor axis (half the distance between the center and the vertices along the y-axis), and a represents the semi-major axis (half the distance between the center and the vertices along the x-axis).
By substituting the values of a (3) and b (2) into the formula, we can find the slopes of the asymptotes:
m = ± 2/3.
Therefore, the slopes of the asymptotes of the hyperbola (x^2)/9 - (y^2)/4 = 1 are ± 2/3.