how would you solve this;
Write an equation for the hyperbola with vertices (-10,1) and (6,1) and foci (-12,1)and (8,1)
The centre is the midpoint between the vertices, which is ((-10+6)/2,(1+1)/2) or (-2,1)
so the equation looks like
(x+2)^2/a^2 - (y-1)^2/b^2 = 1
a is the distance from the centre to the vertex, so a = 8
c is the distance between the centre and the focal point, so c = 10
In a hyperbola a^2 + b^2 = c^2
so 64 + b^2 = 100
b^2 = 36
your equation would be
(x+2)^2/64 - (y-1)^2/36 = 1
To summarize, the equation for the hyperbola with vertices (-10,1) and (6,1) and foci (-12,1) and (8,1) is:
(x+2)^2/64 - (y-1)^2/36 = 1
To solve this problem, you need to understand the properties of a hyperbola and how to determine its equation given certain information. Here is a step-by-step explanation of how to solve the problem and find the equation for the hyperbola:
1. Identify the given information: The vertices of the hyperbola are (-10,1) and (6,1), and the foci are (-12,1) and (8,1).
2. Find the center of the hyperbola: The center is the midpoint between the vertices. Use the midpoint formula: ((x1 + x2)/2, (y1 + y2)/2). In this case, the midpoint is ((-10 + 6)/2, (1 + 1)/2), which simplifies to (-2, 1). Therefore, the center of the hyperbola is (-2, 1).
3. Determine the equation of the hyperbola: The equation of a hyperbola in standard form is given by ((x - h)^2/a^2) - ((y - k)^2/b^2) = 1, where (h,k) is the center of the hyperbola, a is the distance from the center to the vertex, and b is the distance from the center to the endpoint of the conjugate axis.
4. Calculate the value of a: The distance from the center to the vertex is a. In this case, the distance from the center (-2,1) to the vertex (-10,1) is 8 units. Therefore, a = 8.
5. Calculate the value of c: The distance from the center to the focus is c. In this case, the distance from the center (-2,1) to the focus (-12,1) is 10 units. Therefore, c = 10.
6. Use the relationship between a, b, and c: In a hyperbola, the relationship between a, b, and c is a^2 + b^2 = c^2. Plug in the values of a and c to find b^2: 8^2 + b^2 = 10^2. Simplify the equation: 64 + b^2 = 100. Subtract 64 from both sides: b^2 = 36. Take the square root of both sides to find b: b = ±6.
7. Substitute the values into the equation of the hyperbola: Substituting the values of the center (-2,1), a = 8, and b = 6 into the standard form equation ((x - h)^2/a^2) - ((y - k)^2/b^2) = 1, we get ((x + 2)^2/64) - ((y - 1)^2/36) = 1.
Therefore, the equation for the hyperbola with the given information is (x + 2)^2/64 - (y - 1)^2/36 = 1.