Write the equation of the hyperbola with centre at (3,-1),vertex)6,-1),one asymptote with equation2x-4y=9
To write the equation of the hyperbola with the given information, we can follow these steps:
Step 1: Find the center of the hyperbola: The center of the hyperbola is given as (3, -1).
Step 2: Find the coordinates of the vertices: The vertex is given as (6, -1).
Step 3: Find the slope of the asymptote: The asymptote is given by the equation 2x - 4y = 9. To find the slope of the asymptote, we need to re-arrange the equation in the slope-intercept form (y = mx + b).
2x - 4y = 9
-4y = -2x + 9
y = (1/2)x - (9/4)
Comparing with the slope-intercept form (y = mx + b), we can see that the slope of the asymptote is (1/2).
Step 4: Determine the horizontal or vertical hyperbola: Since the given information does not specify whether the hyperbola is horizontal or vertical, we need to establish this based on the slope of the asymptote.
If the slope is positive, the hyperbola is vertical, and if the slope is negative, the hyperbola is horizontal. In this case, the slope of the asymptote is positive (1/2), so the hyperbola is vertical.
Step 5: Write the equation of the hyperbola using the given information:
For a vertical hyperbola with center (h, k), the standard form of the equation is:
(x - h)^2 (y - k)^2
-------- - --------- = 1
a^2 b^2
Where:
(h, k) is the center of the hyperbola.
a is the distance from the center to the vertex.
b is the distance from the center to the foci.
Since we have the center (3, -1) and the vertex (6, -1), we can find the value of a:
a = (distance from center to vertex) = 6 - 3 = 3
Now, we have:
(x - 3)^2 (y + 1)^2
--------- - --------- = 1
3^2 b^2
The value of b can be determined from the slope of the asymptote:
b = |1/2a| = |1/(2*3)| = 1/6
Plug in the values of a and b to the equation:
(x - 3)^2 (y + 1)^2
--------- - --------- = 1
3^2 (1/6)^2
Simplifying further:
(x - 3)^2 36 * (y + 1)^2
--------- - -------------- = 1
9 1
Expanding and rewriting:
36(x - 3)^2
---------- - (y + 1)^2 = 1
9
Multiplying both sides by 9:
4(x - 3)^2 - 9(y + 1)^2 = 36
Therefore, the equation of the hyperbola is:
4(x - 3)^2 - 9(y + 1)^2 = 36