I have to write the equation of a hyperbola whose center is (0,0) and which has a vertical transverse axis.
The equations of the asymptotes are 6x+2y=0 and 6x-2y=0.
I don't even know where to begin. Could someone help? Thanks.
first of all I don't see why the asymptotes were not reduced to
3x ± y = 0
work it backwards.
Recall that for
x^2/a^2 - y^2/b^2 = 1 (the general hyperbola)
the asymptotes are bx ± ay = 0
so b=3 and a=1
for a the equation
x^2 - y^2/9 = 1 or
9x^2 - y^2 = 9
To find the equation of a hyperbola with a vertical transverse axis and given asymptotes, you can follow these steps:
1. Given that the center of the hyperbola is (0,0) and the transverse axis is vertical, we can write the general equation of the hyperbola as:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Since the center is (0,0), we have:
x^2 / a^2 - y^2 / b^2 = 1
2. The slopes of the asymptotes are given by the coefficients of x and y in the equations of the asymptotes. So, the slopes of the asymptotes are -6/2 = -3 and 6/2 = 3.
3. Since the transverse axis is vertical, the equation of the asymptotes takes the form:
y = mx
where m is the slope of the asymptote. We can rewrite the given asymptote equations as:
y = -3x/2 (equation 1)
y = 3x/2 (equation 2)
4. To find the distance between the center (0,0) and a point (x, y) on the hyperbola, we can use the distance formula:
√((x - 0)^2 + (y - 0)^2) = √(x^2 + y^2)
5. For a hyperbola, the distance between the center and a point on the hyperbola is related to the semi-major axis (a) and the semi-minor axis (b) as:
√(x^2 + y^2) = √(a^2 + b^2)
6. Squaring both sides of equation 5, we have:
x^2 + y^2 = a^2 + b^2 (equation 3)
7. We also know that for a hyperbola, the slopes of the asymptotes are related to the semi-major axis (a) and the semi-minor axis (b) as:
slope of asymptotes = ± b / a
8. Substituting the slopes we found in step 3 into the above formula, we get two equations:
-3 = ±b / a (equation 4)
3 = ±b / a (equation 5)
9. Solving equations 4 and 5 simultaneously will give us the values of a and b.
10. Once you have the values of a and b, substitute them into equation 3 to obtain the equation of the hyperbola.
I hope this helps!