Can someone please help...
a hyperbola has vertices (+-5,0) and one focus (6,0) what is the standard form equation of the hyperbola.
The equation for this porblem is
(x-25)-(y/11)=1
Absolutely! I'd be happy to assist you in finding the standard form equation of the hyperbola. Here's how you can do it:
Step 1: Understand the components of a hyperbola:
In order to find the standard form equation of a hyperbola, we need to know a few key elements: the coordinates of the vertices, the coordinates of the foci, and the transverse axis.
Step 2: Determine the center of the hyperbola:
The center of the hyperbola is the midpoint between the vertices. In this case, the x-coordinate of the center is the average of the x-coordinates of the vertices, which is (5 + (-5))/2 = 0. The y-coordinate of the center is 0 since the hyperbola is symmetric around the x-axis.
Step 3: Identify the transverse axis:
The transverse axis is the line segment passing through the center and the vertices. Since the vertices are (+-5,0), the transverse axis lies along the x-axis.
Step 4: Find the distance from the center to the foci:
The distance from the center to each focus is given by the formula c = √(a² + b²), where a represents the distance from the center to the vertex, and b is the distance from the center to a point on the hyperbola.
In this case, the distance from the center (0,0) to the vertex (+5,0) is a = 5. Since the hyperbola is centered at (0,0), b = 0. Therefore, c = √(5² + 0²) = 5.
Step 5: Determine the value of a:
Since the transverse axis is along the x-axis, the value of a is the absolute value of the x-coordinate of the vertex, which is 5.
Step 6: Use the information to write the standard form equation:
The standard form equation of a hyperbola with center (h, k), transverse axis along the x-axis, and distance from the center to the foci c is given by:
(x - h)²/a² - (y - k)²/b² = 1,
where (h, k) represents the coordinates of the center.
Using the information we determined earlier, the standard form equation of the hyperbola is:
(x - 0)²/5² - (y - 0)²/b² = 1,
which simplifies to:
x²/25 - y²/b² = 1.
So, the standard form equation of the hyperbola is x²/25 - y²/b² = 1, where b is the distance from the center to a point on the hyperbola.