Classify the conic section as a circle, an ellipse, a hyperbola, or a parabola.

25x2=64+16y

ellipse

parabola

circle

hyperbola

Parabola

25 x ^ 2 = 64 + 16 y

25 x ^ 2 - 64 = 16 y Divide both sides by 16

25 x ^ 2 / 16 - 64 / 16 = y

y = 25 x ^ 2 / 16 - 4

Quadratic equation.

A parabola is an equation of the form y = a x ^ 2 + b x + c

To classify the conic section, we can use the standard form equations and analyze the coefficients of the variables.

The given equation is 25x^2 = 64 + 16y.

In standard form, the equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.

In standard form, the equation of an ellipse is (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h, k) represents the center of the ellipse and a and b represent the semi-major and semi-minor axes.

In standard form, the equation of a hyperbola is (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1, where (h, k) represents the center of the hyperbola, and a and b represent the distances from the center to the vertices.

In standard form, the equation of a parabola is (x - h)^2 = 4a(y - k), where (h, k) represents the vertex of the parabola, and a represents the distance between the vertex and focus.

Comparing the given equation, 25x^2 = 64 + 16y, to the standard form equations, we can see that there is no term present for x^2 or y^2. This implies that the equation represents a type of conic section where either the coefficient of x^2 or y^2 is zero. In this case, the coefficient of y^2 is zero.

Therefore, the given equation represents a parabola.