x^2-y^2-6x+8y-3=0
Find the center vertices, foci, and asymptotes for the hyperbola.
x^2-y^2-6x+8y-3=0
x^2 - 6x + 9 - (y^2 - 8y + 16) = 3 + 9 + -16
(x-3)^2 - (y-4)^2 = -4
(x-3)^2/4 - (y-4)^2/4 = -1
You should be able to take it from there
To find the center, vertices, foci, and asymptotes of a hyperbola, we need to first rewrite the given equation in the standard form. The standard form of a hyperbola equation is:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1 [for a horizontal hyperbola]
or
((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1 [for a vertical hyperbola]
Comparing this with the given equation x^2 - y^2 - 6x + 8y - 3 = 0, we can see that it can be rewritten by completing the square for both x and y terms.
x^2 - 6x - y^2 + 8y = 3
Rearranging the terms, we get:
(x^2 - 6x) - (y^2 - 8y) = 3
Next, we group the x and y terms separately and complete the square within each group.
(x^2 - 6x + 9) - (y^2 - 8y + 16) = 3 + 9 - 16
(x - 3)^2 - (y - 4)^2 = -4
Now, divide by -4 to get the equation in the standard form:
((x - 3)^2 / 4) - ((y - 4)^2 / -4) = 1
From this equation, we can determine the necessary information about the hyperbola:
1. Center: The center of the hyperbola is (h, k), which is (3, 4) in this case.
2. Vertices: The distance from the center to the vertices is given by a, and it can be calculated as the square root of the denominator of the positive term of the x-coordinate. In this case, a = √4 = 2. Therefore, the vertices are located at (3 ± 2, 4), which gives us two points (1, 4) and (5, 4).
3. Foci: The distance from the center to the foci is given by c, and it can be calculated as the square root of a^2 + b^2. In this case, since a = 2, we need to determine b. Using the formula b^2 = a^2 + c^2, we can substitute a = 2 and solve for b^2. Assuming c^2 = 1 (because it's positive for a horizontal hyperbola), we get b^2 = 4 + 1 = 5. Taking the square root, we find b ≈ √5. Therefore, c = √(2^2 + √5^2) = √9 = 3. Thus, the foci are located at (3 ± 3, 4), giving us two points (0, 4) and (6, 4).
4. Asymptotes: The equations of the asymptotes can be determined using the formula y = ± (b/a) * (x - h) + k. Substituting the known values, we get:
y = ± (√5/2) * (x - 3) + 4
Therefore, the equations of the asymptotes are:
y = (√5/2)x - (√5/2) + 4
y = - (√5/2)x + (√5/2) + 4
In summary, the information about the hyperbola is:
- Center: (3, 4)
- Vertices: (1, 4) and (5, 4)
- Foci: (0, 4) and (6, 4)
- Asymptotes: y = (√5/2)x - (√5/2) + 4 and y = - (√5/2)x + (√5/2) + 4