 # 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 avissidep 2020-12-24 Answered
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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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 ${x}^{2}+{y}^{2}=4\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{x}^{2}+{z}^{2}=4$
###### Not exactly what you’re looking for? 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.

###### Not exactly what you’re looking for? Jeffrey Jordon