Find the standard form of the equation of the hyperbola with the given characteristics.
foci: (±4, 0) asymptotes: y= +/- 3x
we know c=4 and b/a = 3/1 so b = 3a
also in a standard hyperbola
a^2 + b^2 = c^2
a^2 + (9a^2) = 16
a^2 = 16/10 = 8/5
b^2 = 9a^2 = 72/5
standard form:
x^2/((8/5) - y^2/((72/5)) = 1
Thank you!!I was just having a hard time on finding out a and b but now I understand.
To find the standard form of the equation of a hyperbola, we need to determine the coordinates of the center, the distance between the center and the foci, and the direction of the transverse axis.
Since the foci are located at (±4, 0), we can determine that the center of the hyperbola is at the origin (0, 0).
The asymptotes of the hyperbola have slopes of ±3, indicating that the hyperbola is vertically oriented.
To determine the distance between the center and the foci (c), we can use the formula:
c = sqrt(a^2 + b^2), where c is the distance between the center and the foci and a is the distance between the center and the vertices.
Since the slopes of the asymptotes are ±3, the value of a/b (where a represents the semi-major axis and b represents the semi-minor axis) is 3.
We also know that the distance between the center and each focus is 4, so we have:
c = sqrt(3^2 + b^2) = 4.
Solving for b gives us:
9 + b^2 = 16
b^2 = 16 - 9
b^2 = 7
b = sqrt(7)
Now that we have the values of a and b, we can write the standard form of the equation of the hyperbola:
For a vertically oriented hyperbola:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1.
Plugging in the values, we have:
(x - 0)^2 / 3^2 - (y - 0)^2 / (sqrt(7))^2 = 1
Simplifying, we get:
x^2 / 9 - y^2 / 7 = 1
Therefore, the standard form of the equation of the hyperbola with the given characteristics is:
x^2 / 9 - y^2 / 7 = 1.
To find the standard form of the equation of a hyperbola, you need to know the coordinates of the foci and the equations of the asymptotes.
In this case, the foci are at (±4, 0) and the asymptotes have equations y = ±3x.
The standard form of the equation of a hyperbola is given by:
(x - h)²/a² - (y - k)²/b² = 1 or
(y - k)²/a² - (x - h)²/b² = 1
To determine which form to use, we need to determine the orientation of the hyperbola based on the given characteristics.
Since the equation of the asymptotes is in the form y = mx, where m is the slope, we can determine the orientation of the hyperbola from the ratio of the coefficients. If the coefficient of x is positive, the hyperbola is oriented vertically (bigger along the y-axis), and if it is negative, the hyperbola is oriented horizontally (bigger along the x-axis).
In this case, since the coefficient of x in the equation of the asymptotes is positive (3x), the hyperbola is oriented vertically.
Given that the foci are (±4, 0), the center of the hyperbola lies at the origin (0, 0). So, h = k = 0 in the standard form equation.
Now, let's focus on finding the values of a and b.
For a vertical hyperbola, a represents the distance from the center to the vertex, and b represents the distance from the center to the asymptotes.
To find a, we can start by finding the distance from the center to each focus. The distance formula is given by:
d = √((x2 - x1)² + (y2 - y1)²)
Using the coordinates of one focus (±4, 0) and the center (0, 0), we can compute the distance:
d = √((0 - 4)² + (0 - 0)²) = √16 = 4
Therefore, a = 4.
To find b, we can consider the slope of one asymptote, which is given as 3. The value of b can be computed using the formula:
b = a/c, where c is the distance from the center to one focus.
We already know that a = 4, and we can compute c using the distance formula:
c = √((0 - 4)² + (0 - 0)²) = √16 = 4
Therefore, b = a/c = 4/4 = 1.
Since the hyperbola is oriented vertically, the standard form of the equation is:
(x - h)²/a² - (y - k)²/b² = 1
Plugging in the known values, we have:
(x - 0)²/4² - (y - 0)²/1² = 1
Simplifying, we get:
x²/16 - y² = 1
So, the standard form of the equation of the hyperbola is x²/16 - y² = 1.