Identify the graph of the equation 4x^2-25y^2=100. Then write the equation of the translated graph for T(5,-2) in general form.
Might be another case of half right=all wrong.
I left off that it's a hyperbola
(x-h)^2/a^2 - (y-k)^2/b^2 = 1
x^2/25 - y^2/4 = 1
The given equation, 4x^2 - 25y^2 = 100, represents a hyperbola. To identify the graph, we can rewrite the equation in the general form of a hyperbola:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Comparing this general form with the given equation, we can determine the values of h, k, a, and b.
Comparing the x terms:
a^2 = 25
a = 5 (since a cannot be negative)
Comparing the y terms:
b^2 = 4
b = 2 (since b cannot be negative)
Therefore, the general form of the translated graph for T(5, -2) is:
(x - 5)^2 / 25 - (y + 2)^2 / 4 = 1
To identify the graph of the given equation and write it in general form, we need to determine the type of conic section and its center.
The equation provided, 4x^2 - 25y^2 = 100, is a hyperbola. To write it in general form, we need to rearrange the equation to match the standard equation of a hyperbola:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Comparing this standard form with the given equation, we can see that a^2 = 25 and b^2 = 4. Therefore, a = 5 and b = 2.
Now, let's determine the center of the hyperbola:
The center of a hyperbola in the general form (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1 is denoted by the coordinates (h, k).
In this case, we can rewrite the given equation as:
(x - 0)^2 / 5^2 - (y - 0)^2 / 2^2 = 1
By comparing the two equations, we can deduce that the center of the hyperbola is at the point (0, 0).
So, the graph of the equation 4x^2 - 25y^2 = 100 is a hyperbola centered at the origin (0, 0), with the horizontal axis having a length of 2a = 10 (since a = 5), and the vertical axis having a length of 2b = 4 (since b = 2).
Now, to find the equation of the translated graph for T(5, -2) in general form, we need to shift the center from (0, 0) to (5, -2).
The new equation in general form can be written as:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Substituting the values h = 5, k = -2, a = 5, and b = 2, into the general form equation, we have:
(x - 5)^2 / 5^2 - (y - (-2))^2 / 2^2 = 1
Simplifying this equation will give you the general form of the translated hyperbola.