Question

Give a geometric description of the set of points $(x,y,z)$ satisfying the pair of equations $z=0$ and $x^2+y^2=1.$ Sketch a figure of this set of points.

Short answer

Answer: The geometric description is a circle with a radius of 1, lying on the XY plane (z=0).

Step-by-step solution

Analyze the equation $z=0$

The equation $z=0$ corresponds to the plane parallel to the XY plane and passing through the origin (0,0,0). This should be considered when visualizing the geometric representation of the solution.

Analyze the equation $x^2+y^2=1$

The equation $x^2+y^2=1$ represents a circle with radius 1 in the XY plane, centered at the origin (0,0).

Combine observations

Since both $z=0$ and $x^2+y^2=1$ should be satisfied simultaneously, the set of points must lie on the plane $z=0$. It is also required that the points should lie on the circle $x^2+y^2=1$. Therefore, the geometric description of the set of points is a circle with a radius of 1, lying on the XY plane.

Sketch the figure

To sketch this geometric description, draw the XY plane, then draw the circle with a radius of 1 centered at the origin (0, 0). The circle will be in the XY plane with z=0. This circle represents the set of points $(x,y,z)$ that satisfy the given equations.

Key concepts

Geometric Representation

When tackling problems in 3D coordinate geometry, the geometric representation of solutions is crucial for understanding and visualization. In the given exercise, two equations are defining a set of points. The first equation, z=0, signifies a plane that is parallel to the XY plane. This means that all points on this plane have a z-coordinate of zero, effectively making this a 2D plane within the 3D space.

The second equation, x^2 + y^2 = 1, defines a circle with a radius of 1, but because no z-coordinate is specified, this circle could theoretically be at any height. It's the combination of these two equations that restricts the circle to the XY plane. Representing this graphically is a powerful tool for students to intuitively grasp how these two equations intersect to produce a simple, yet elegant, geometric figure: a flat circle.

Circle in XY Plane

The equation x^2 + y^2 = 1 is a classic representation of a circle in 2D space, specifically the XY plane. It indicates that for any point on the circle, the sum of the squares of the x and y coordinates is equal to 1. This circle is centered at the origin with the coordinates (0, 0) and has a radius equal to 1. An understanding of this is fundamental to visualizing the shape in 3D space.

Considering the additional constraint of z=0, it is clear that the circle is locked into the XY plane within the 3D coordinate system. This highlights an essential concept of 3D coordinate geometry; shapes can be constrained to two dimensions even within a three-dimensional space, often through the use of equations that limit one of the coordinates—in this case, the z-coordinate.

Geometric Sketching

Visualizing complex geometric concepts often becomes easier through sketching. The act of geometric sketching bridges the gap between abstract equations and tangible representations. For the set of points defined by z=0 and x^2 + y^2 = 1, creating a sketch helps students to visualize the 2D figure within the context of 3D space.

To sketch the set of points satisfying the given equations, start by drawing the XY plane. Then, within this plane, draw a circle centered at the origin (0,0) with a radius of 1. It may help to mark several points along the circle to ensure its shape is accurate and to reinforce that these points all satisfy the equation x^2 + y^2 = 1. This sketch serves as a critical thinking aid, allowing students to move from theoretical understanding to practical application.

Equations of a Circle

Understanding the equations of a circle is pivotal when interpreting or sketching geometric figures in coordinate systems. The standard form of the equation for a circle in two dimensions is given by (x-h)^2 + (y-k)^2 = r^2, which describes a circle with a center at (h, k) and a radius of r. In the context of the given problem, the equation simplifies to x^2 + y^2 = 1 because the circle is centered at the origin, making h and k equal to zero, and the radius r equal to 1.

It's crucial to comprehend that the simple form of this equation signifies a specific set of conditions: the circle's center at the origin and its size being constrained by the radius. Any deviation from these conditions would lead to a more complex equation representing circles with different radii or centers located at different points in the plane.