okay so the question is as follows:
determien the standard form of the equation of a hyperbola with vertices(+or- 2,0) and passing through (4,3)
is this right?
x^2/2^2 - y^2/b^2 =1
Since P(4,3) passes through, the x and y coordinates satisfy the equation:
4^2/2^2 - 3^2/b^2 =1
b^2=3
thus the equation is
x^2/4 -y^2/3=1
is this right?
Yes, your solution is correct.
To find the standard form of the equation of a hyperbola, you need to use the given information about the vertices and a point on the hyperbola. Here's how you can determine the equation:
1. The standard form of the equation of a hyperbola is (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h,k) represents the center of the hyperbola.
2. Since the given vertices are (+/-2, 0), the center is at (h, k) = (0, 0). Therefore, the equation becomes x^2/a^2 - y^2/b^2 = 1.
3. To determine the value of a, we know that the distance from the center to each vertex is a. In this case, |a| = 2, so a = 2.
4. Now we have x^2/2^2 - y^2/b^2 = 1.
5. Finally, we are given that the hyperbola passes through the point (4, 3). Substituting these coordinates into the equation, we get 4^2/2^2 - 3^2/b^2 = 1.
6. Solving for b^2, we find b^2 = 3.
7. Substituting a = 2 and b^2 = 3 back into the equation, we arrive at the final equation: x^2/4 - y^2/3 = 1.
So, your solution x^2/4 - y^2/3 = 1 is correct.