Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. X 2 + y 2 = 4, z = y
Answers
Answer:
An ellipse where the plane z=y cuts the cylinder x^2 + y^2 = 4.
Step-by-step explanation:
The points that satisfy x^2 + y^2 = 4 in the xy-plane are the points of a circle with radius one centred at the origin. In the xy-plane, we have z=0, but as z plays no role here, every point above or below the points of this circle will satisfy this equation, too. Consequently, the points that satisfy x^2 + y^2 = 4 are the points of an infinitely long cylinder around the z-axis with radius 2.
In the yz-plane, the equation z=y describes a line through the origin at 45 degrees to the y-axis. As before, the x coordinate is free to be whatever it likes, to in space, the equation z = y describes the plane through the x-axis that make a 45 degree angle to the y-axis (and so also to the z-axis).
The points that satisfy both equations simultaneously are then the points on both of these surfaces. So imagine the cylinder around the z-axis, standing tall, and the 45 degree angle plane cutting through that cylinder. The points of intersection form an ellipse. This is the set of points whose coordinates satisfy both equations.