(-2,1) and (2,1)
Passes through (5,4)
What is the standard form of the equation of the hyperbola?
(-2,1) and (2,1) are the vertices.
Wouldn't it be:
x^2/4 - (y-1)^2/(9/5)=1
because that's what you get plug b^2 back into the original equation?
5(y-1)^2/9 is the same as (y-1)^2/(9/5)
I don't like double decker fractions
To determine the standard form of the equation of a hyperbola, you will need the coordinates of the center, the vertices, and the foci of the hyperbola. Given that the hyperbola passes through the point (5,4) and its vertices are (-2,1) and (2,1), we can follow these steps to find the equation:
Step 1: Find the center of the hyperbola.
The center of a hyperbola is the midpoint of the line segment joining the two vertices. In this case, the x-coordinate of the center is the average of the x-coordinates of the vertices, and the y-coordinate of the center remains the same. Thus, the center of the hyperbola is (0,1).
Step 2: Find the distance between the center and one of the vertices.
By using the distance formula, calculate the distance between the center (0,1) and one of the vertices. In this case, choose either (-2,1) or (2,1). Let's use (-2,1) as an example.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((-2 - 0)^2 + (1 - 1)^2)
Distance = sqrt(4 + 0)
Distance = sqrt(4)
Distance = 2
So, the distance between the center and one of the vertices is 2.
Step 3: Using the distance from Step 2, find the value of "a."
In the standard form of the equation of a hyperbola, a represents the distance from the center to either vertex. Therefore, a = 2.
Step 4: Find the value of "b."
The distance between the center of a hyperbola and one of its foci is given by the formula c = sqrt(a^2 + b^2), where c represents the distance between the center and a focus. Since the hyperbola passes through the point (5,4), you can use this information to find the value of c.
Distance from (5,4) to the center (0,1):
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((5 - 0)^2 + (4 - 1)^2)
Distance = sqrt(25 + 9)
Distance = sqrt(34)
Since c is the distance between the center and a focus, and a = 2 from Step 3, we can solve for b using the equation c = sqrt(a^2 + b^2):
sqrt(34) = sqrt(2^2 + b^2)
34 = 4 + b^2
b^2 = 30
b = sqrt(30)
So, the value of b is sqrt(30).
Step 5: Write the equation in standard form.
The standard form of the equation of a hyperbola with center (h, k), and values of a and b, is ((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1. Plug in the values we've determined to get the final equation:
((x - 0)^2 / 2^2) - ((y - 1)^2 / (sqrt(30))^2) = 1
(x^2 / 4) - ((y - 1)^2 / 30) = 1
Therefore, the standard form of the equation of the hyperbola is (x^2 / 4) - ((y - 1)^2 / 30) = 1.
the centre is clearly (0,1), the midpoint of these two vertices.
and a = 2
so x^2/4 - (y-1)^2/b^2 = 1
but it passes through the point (5,4), so
25/4 - 9/b^2 = 1
-9/b^2= 1-24/4
-9/b^2 = -20/4
b^2/9 = 4/20 = 1/5
b^2 = 9/5
so the hyperbola is
x^2/25 - 5(y-1)^2/9 = 1
(check my arithmetic)