Chapter 8.1, Problem 73E

Solution

a.

To find: What two lines form the graph of the solution set.

The two lines $y=x+1$ and $y=x-1$ form the graph of the solution set.

Given information:

The given equations are,

  $\begin{array}{c}{x}^2-2xy+y^2-1=0\\ \left(x-y+1\right)\left(x-y-1\right)=0\end{array}$

Calculation:

The given equations are,

  $\begin{array}{c}{x}^2-2xy+y^2-1=0\\ \left(x-y+1\right)\left(x-y-1\right)=0\end{array}$

Comparing both the equations,

  $x^2-2xy+y^2-1=\left(x-y+1\right)\left(x-y-1\right)$

Take each factor equal to $0$

  $\begin{array}{c}x-y+1=0\\ y=x+1\\ x-y-1=0\\ y=x-1\end{array}$

Therefore, the two lines $y=x+1$ and $y=x-1$ form the graph of the solution set.

b.

To prove: Why the graph in (a) not covered by the case in the given figure.

Given information:

The given equations are,

  $\begin{array}{c}{x}^2-2xy+y^2-1=0\\ \left(x-y+1\right)\left(x-y-1\right)=0\end{array}$

Proof:

From part (a) the graph plotted is,

  

Interpretation:

From the graph it shows that,

The lines are parallel, not intersecting.

The cone in Figure $8.2$ cannot be intersected by a plane to produce two parallel lines.

c.

To find: The points $\left(x,y\right)$ solve the equation $x^2+y^2+1=0$ .

There are no points to solve the equation $x^2+y^2+1=0$ .

Given information:

The given equation is,

  $x^2+y^2+1=0$

Calculation:

From the graph it is noted that,

Since $x^2+y^2$ cannot be negative, there are no positions $(x,y)$ that solve the given equation.

Therefore, there are no points $(x,y)$ that solve the equation $x^2+y^2+1=0$ .

d.

To prove: The graph in (c) not covered by the case in the provide figure.

Given information:

The given equations are,

  $x^2+y^2+1=0$

Calculation:

From Figure $8.2$ a cone that is indefinitely long and intersects every possible plane in a nonempty way.

Therefore, the graph would be the empty set.

e.

To prove: How each of the graphs in parts (a) and (c) can be realized as the intersection of the cylinder with an appropriate plane by drawing a cylinder.

Given information:

Consider an infinitely extended cylinder to be a degenerate “cone” with a vertex at infinity.

Proof:

Let’s consider an infinitely extended cylinder to be a degenerate “cone” with a vertex at infinity.

  

  

Therefore, the graphs in parts (a) and (c) can be realized as the intersection of the cylinder with two parallel lines and an empty set.