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
x^2/25 - y^2/4 = 1 is the correct simplification of the original hyperbola
now according to your transformation
T(5,-2) I will asssume that you mean that
(x,y) ----> (x+5,y-2)
so the original centre of (0,0) would move to (5,-2) and your new equation is
[(x-5)^2]/25 - [(y+2)^2]/4 = 1
To identify the graph of the equation 4x^2 - 25y^2 = 100, we can start by re-arranging the equation to the standard form of a hyperbola. The standard form for a hyperbola with a horizontal transverse axis is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Comparing this with the given equation, we can determine the values of a, b, h, and k. In this case, the equation is already in standard form, so we can directly identify the values:
a^2 = 25
b^2 = 4
h = 0 (since there is no x term with a coefficient other than 1)
k = 0 (since there is no y term with a coefficient other than 1)
From these values, we can conclude that the center of the hyperbola is at the origin (0,0), the transverse axis is 2a = 2√25 = 10 units long in the x-direction, and the conjugate axis is 2b = 2√4 = 4 units long in the y-direction.
Now, let's write the equation of the translated graph for T(5,-2) in general form. To translate the graph, we need to shift the center from (0,0) to (5,-2). The general form of the equation for the translated hyperbola is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Substituting the values of h = 5 and k = -2, we get:
(x - 5)^2 / a^2 - (y + 2)^2 / b^2 = 1
Since we already know a^2 = 25 and b^2 = 4, the final equation in general form becomes:
(x - 5)^2 / 25 - (y + 2)^2 / 4 = 1