Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. x2+y2=4,z=y

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stuth1 · Accepted Answer

Step 1  Given:  The pair of equations:  x2+y2=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 x2+y2=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 x2+y2=4 which satisfy the two circle x2+y2=4andx2+z2=4

Jeffrey Jordon · Accepted Answer

Answer:  An ellipse where the plane z=y cuts the cylinder x2+y2=4.  Step-by-step explanation:  The points that satisfy x2+y2=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 x2+y2=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 · Accepted Answer

Answer is given below (on video)

Eliza Beth13 · Accepted Answer

Answer:The set of points in space that satisfy the given pair of equations x2+y2=4 and z=y forms a circular cylinder with a radius of 2, centered at the origin (0,0,0), and infinite height along the z-axis.Explanation:Let&#039;s start with the first equation, x2+y2=4. This equation represents a circle in the xy-plane centered at the origin (0,0) with a radius of 2. The equation can be rewritten in polar coordinates as r=2, where r represents the distance from the origin to a point (x,y). Therefore, all points (x,y) that lie on the circle of radius 2 centered at the origin satisfy this equation.Next, let&#039;s consider the second equation, z=y. This equation states that the z-coordinate is equal to the y-coordinate. Geometrically, this means that the points lie on a plane parallel to the xz-plane, and the z-coordinate is equal to the y-coordinate. In other words, if we fix any value of z, the corresponding y-coordinate will be the same.Now, let&#039;s combine the two equations to find the solution for the set of points in space that satisfy both equations. Since z=y, we can substitute y for z in the equation of the circle: x2+z2=4. This equation represents a circular cylinder in 3D space with a radius of 2 and infinite height along the z-axis. The center of the circular base is still at the origin (0,0,0), but the points on the cylinder extend along the z-axis indefinitely.

Nick Camelot · Accepted Answer

The geometric description of the set of points in space satisfying the given equations x2+y2=4 and z=y is a cylinder with radius 2 centered along the x-axis, where the height of each point on the cylinder is equal to its y-coordinate.

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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