Hyperbolas:
Could someone please tell me how to determine if the following indicate the foci are on the x-axis or the y-axis?
x^2/4 -y^2/16 = 1
x^2/49 - y^2/9 = 1
x^2/36 - y^2/16 = 1
please explain how to set the equations up
thank you
These are all hyperbolas.
Look at wwhat happens when y = 0 (on the x axis) and when x = 0 (on the y axis)
In all cases, when y = 0, there are two real solutions for x. In the first one for example when y = 0, x = 2 or x = -2
However when x = 0, there is no real solution for y. In other words the hyperbolas never reach the y axis and open left and right. All foci are therefore located on the x axis.
To determine whether the foci are on the x-axis or the y-axis in a hyperbola, you can examine the equations. Here's how you can set up the equations and analyze them:
1. x^2/4 - y^2/16 = 1:
To determine the orientation of the hyperbola, we need to look at the denominators of the x^2 and y^2 terms. If the denominator of the x^2 term is larger than the denominator of the y^2 term, the foci are on the y-axis, and vice versa.
In this equation, the denominator of the x^2 term is 4, and the denominator of the y^2 term is 16. Since 16 > 4, the foci are on the y-axis.
2. x^2/49 - y^2/9 = 1:
Similarly, comparing the denominators, we find that the denominator of the x^2 term is 49, while the denominator of the y^2 term is 9. Again, since 49 > 9, the foci are on the y-axis.
3. x^2/36 - y^2/16 = 1:
For this equation, the denominator of the x^2 term is 36, and the denominator of the y^2 term is 16. Since 16 < 36, the foci are on the x-axis.
So, in summary:
- For the first two equations, the foci are on the y-axis.
- For the third equation, the foci are on the x-axis.
To determine whether the foci of a hyperbola are on the x-axis or y-axis, we need to examine the coefficients of the x² and y² terms in the given equations.
A hyperbola with the foci on the x-axis has the form: (x - h)²/a² - (y - k)²/b² = 1, where a represents the distance from the center to a vertex along the x-axis.
A hyperbola with the foci on the y-axis has the form: (y - k)²/a² - (x - h)²/b² = 1, where a represents the distance from the center to a vertex along the y-axis.
Now let's analyze the equations you provided:
1. x²/4 - y²/16 = 1
The coefficient of x² is 1/4, and the coefficient of y² is -1/16. Since the coefficient of x² is positive and greater than the coefficient of y² (which is negative), we can conclude that the foci of this hyperbola are on the x-axis.
2. x²/49 - y²/9 = 1
The coefficient of x² is 1/49, and the coefficient of y² is -1/9. Similar to the previous equation, the coefficient of x² is positive, while the coefficient of y² is negative. Hence, the foci of this hyperbola are also on the x-axis.
3. x²/36 - y²/16 = 1
The coefficient of x² is 1/36, and the coefficient of y² is -1/16. Once again, the coefficient of x² is positive, but this time it is smaller than the coefficient of y². Therefore, the foci of this hyperbola are on the y-axis.
By examining the coefficients of the x² and y² terms, we can determine whether the foci of the hyperbolas are on the x-axis or the y-axis.