x^2-4y^2=0 is what conic section?

To identify the conic section described by the equation x^2 - 4y^2 = 0, we need to examine the equation's coefficients. In this case, the equation has a positive coefficient for x^2 and a negative coefficient for y^2, indicating that it is an equation of a hyperbola.

To mathematically prove this, we can rewrite the equation in standard form. Firstly, we divide both sides of the equation by 4 to obtain:

(x^2)/4 - (y^2)/1 = 0

Now, we can rearrange the equation to isolate the two variables by moving the constant value to the other side:

(x^2)/4 = (y^2)/1

Next, we can separate the fractions to better examine the equation:

x^2/4 = y^2/1

Now, observe that the equation on the left side can be written as:

(x/2)^2 = y^2

Comparing this with the standard equation for a hyperbola, we have:

(x/a)^2 - (y/b)^2 = 1

From the comparison, we can see that a = 2 and b = 1. Since the equation has a positive coefficient for x^2 and a negative coefficient for y^2, we can conclude that the given equation represents a hyperbola.