# 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

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. ${x}^{2}+{y}^{2}=4,z=y$
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stuth1
Step 1
Given:
The pair of equations:
${x}^{2}+{y}^{2}=4,z=y$
We have to give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations.
Step 2
The equation ${x}^{2}+{y}^{2}=4$ gives the circle in the X-Y plane with centre of the circle is the origin(0, 0) with radius of the circle is r=2 and the equation z=y is the plane
The pair of both the equation is the intersection of the plane z=y and circle ${x}^{2}+{y}^{2}=4$ which satisfy the two circle ${x}^{2}+{y}^{2}=4\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{x}^{2}+{z}^{2}=4$
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Jeffrey Jordon

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.

Jeffrey Jordon