Assessment: Circles and Parabolas

Which equation represents a parabola?

A circle has equation (x−2)²+(y+3)²=64. Find its radius.

Find the radius of the circle x²+y²−4x+6y−3=0.

For the parabola x²=20y, find p.

A parabola has focus (0,4) and directrix y=−4. Find the distance from the point (8,4) to the focus.

A parabola (x+3)²=12(y−1) has vertex (h,1). Find h.

A surveyor moves their reference origin to a more convenient landmark on the map, then re-measures every point's coordinates relative to it. Which technique describes this?

An engineer analyzing a tilted conveyor belt rotates the coordinate axes to align with the belt's actual direction of motion, simplifying the equations of motion. Which technique is this?

Find the y-coordinate of the center of the circle x²+y²+8x−2y+13=0.

A parabola (x−5)²=−8(y−2) has vertex (5,2) and p=−2 (opens downward). Find the y-coordinate of its focus, which is at (h, k+p).