Final answer:
The equation x²=4y−2y² is classified as B) parabola after rearranging it to the standard form and analyzing the coefficients of the squared terms, which do not match the criteria of an ellipse, hyperbola, or circle.
Step-by-step explanation:
The equation provided, x²=4y−2y², represents a specific type of conic section. To classify this conic section, we need to rewrite the equation in a standard form that enables us to identify its type easily. By rearranging the equation, we have:
x² + 2y² - 4y = 0
Now let's complete the square for the y terms, by adding 4 (the square of half the coefficient of y) on both sides:
x² + 2(y² - 2y + 1) = 2
This simplifies to:
x² + 2(y - 1)² = 2
y - 1 is squared and multiplied by a positive number, and x is squared and not multiplied by a negative number, indicating this is not a hyperbola. Furthermore, there are no x and y cross terms and coefficients of x² and y² are different, which excludes the possibility of a circle.
Given these attributes and knowing the equation does not represent an ellipse because the x² and y² terms would have to have the same coefficients and opposite signs, we can conclude that this equation represents a parabola. Therefore, the correct option is B) Parabola.