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. x2+y2=4,z=y
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

stuth1
Answered 2020-12-25 Author has 97 answers
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
Not exactly what you’re looking for?
Ask My Question
Jeffrey Jordon
Answered 2021-10-08 Author has 2064 answers

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.

Not exactly what you’re looking for?
Ask My Question
Jeffrey Jordon
Answered 2021-11-08 Author has 2064 answers

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more