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, that is correct! You have correctly determined the standard form of the equation of a hyperbola with the given information.
To explain the steps to get the answer, let's start with the standard form equation of a hyperbola:
(x-h)^2/a^2 - (y-k)^2/b^2 = 1
where (h,k) represents the center of the hyperbola.
First, we are given the vertices (+/- 2, 0), which means the center of the hyperbola is at the origin (0, 0).
So, our equation becomes:
x^2/a^2 - y^2/b^2 = 1
Next, we know that the hyperbola passes through the point (4, 3). We can substitute this point into the equation and solve for 'a' and 'b'.
When we substitute (4, 3) into the equation, we get:
4^2/a^2 - 3^2/b^2 = 1
16/a^2 - 9/b^2 = 1
Since we are given that the vertices are (+/-2, 0), which means the distance 'a' from the center to the vertex is 2, we can substitute 'a = 2' into the equation:
16/2^2 - 9/b^2 = 1
16/4 - 9/b^2 = 1
4 - 9/b^2 = 1
-9/b^2 = -3
To solve for 'b', we can cross-multiply:
b^2 = 3
Now that we have found the value of 'b', we can substitute it back into the equation:
x^2/2^2 - y^2/3 = 1
Simplifying further:
x^2/4 - y^2/3 = 1
Therefore, your final answer is correct:
x^2/4 - y^2/3 = 1