Using the information given below write the equation of the hyperbola:
The transverse axis of the hyperbola lies on the line y=–3 and has length 6; the conjugate axis lies on the line x=2 and has length 8.
transverse axis y = -3
so, (x-h)^2/a^2 - (y+3)^2/b^2 = 1
transverse axis has length 6, so a=3
(x-h)^2/9 - (y+3)^2/b^2 = 1
conjugate axis x=2, so h=2
(x-2)^2/9 - (y+3)^2 = 1
conjugate axis has length 8, so b=4
(x-2)^2/9 - (y+3)^2/16 = 1
wikipedia has a thorough article on hyperbolas.
But I'm sure all this is in your text as well.
Sure, here's the equation of the hyperbola:
Why did the hyperbola go to therapy?
Because it had axis issues!
But seriously, let's get back to mathematics.
The equation of the hyperbola is given by:
$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$
Where (h, k) represents the center of the hyperbola.
From the given information, we can deduce that the center of the hyperbola is (2, -3).
Also, we are given that the length of the transverse axis is 6, which means 2a = 6. Therefore, a = 3.
Similarly, the length of the conjugate axis is 8, which means 2b = 8. Therefore, b = 4.
Plugging these values into the equation, we get:
$\frac{(x-2)^2}{3^2} - \frac{(y+3)^2}{4^2} = 1$
So, the equation of the hyperbola is:
$\frac{(x-2)^2}{9} - \frac{(y+3)^2}{16} = 1$
Hope this mathematical clowning around helps you!
To find the equation of the hyperbola, let's analyze the given information.
1. The transverse axis lies on the line y = -3 and has a length of 6.
The center of the hyperbola will have coordinates (h, k), where h is the x-coordinate and k is the y-coordinate of the center.
Given that the transverse axis lies on the line y = -3, the y-coordinate of the center, k, is -3.
The length of the transverse axis is 6, so the distance from the center to each vertex is a = 6/2 = 3.
2. The conjugate axis lies on the line x = 2 and has a length of 8.
The line x = 2 represents the vertical line on which the conjugate axis lies. The x-coordinate of the center, h, is 2.
The length of the conjugate axis is 8, so the distance from the center to each co-vertex is b = 8/2 = 4.
Now we have the center (h, k) = (2, -3), the distance from the center to each vertex a = 3, and the distance from the center to each co-vertex b = 4.
The equation of a hyperbola with a horizontal transverse axis is given by:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Plugging in the values:
(x - 2)^2 / 3^2 - (y + 3)^2 / 4^2 = 1
Simplifying this equation gives the equation of the hyperbola.
To write the equation of the hyperbola, we need to use the standard form of the equation for a hyperbola. The standard form for a hyperbola with a center at (h, k), a transverse axis length of 2a, and a conjugate axis length of 2b is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Let's break down the given information to determine the values we need for the equation:
1. The transverse axis of the hyperbola lies on the line y = -3 and has a length of 6.
Since the transverse axis is parallel to the x-axis, its equation will have the form y = constant. We are given that y = -3, so the center of the hyperbola (h, k) will have a y-coordinate of -3.
2. The conjugate axis of the hyperbola lies on the line x = 2 and has a length of 8.
Since the conjugate axis is parallel to the y-axis, its equation will have the form x = constant. We are given that x = 2, so the center of the hyperbola (h, k) will have an x-coordinate of 2.
Using the information from steps 1 and 2, we can determine the center of the hyperbola to be (2, -3).
Next, we need to find the values of a and b, which represent half the lengths of the transverse and conjugate axes, respectively. We are given that the transverse axis has a length of 6, so a = 6/2 = 3. The conjugate axis has a length of 8, so b = 8/2 = 4.
Finally, we can plug the values into the standard form of the equation:
(x - 2)^2 / 3^2 - (y + 3)^2 / 4^2 = 1
Simplifying, we get the equation of the hyperbola:
(x - 2)^2 / 9 - (y + 3)^2 / 16 = 1