Chapter 8.3, Problem 72E

Solution

How the given graph is similar to and different to a full-fledged hyperbola.

It has been explained how the given graph is similar to and different to a full-fledged hyperbola.

Given:

The graph of the equation, $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=0$ is considered to be a degenerate hyperbola.

Concept used:

A hyperbola is formed when a plane intersects a double-sided cone in both nappes.

Calculation:

The given equation is $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=0$ .

Simplifying,

  $\frac{{\left(y-k\right)}^2}{b^2}=\frac{{\left(x-h\right)}^2}{a^2}$

On further simplification,

  ${\left(y-k\right)}^2=\frac{b^2}{a^2}{\left(x-h\right)}^2$

Continuing simplification,

  $y-k=\pm \frac{b}{a}\left(x-h\right)$

Finally,

  $y=\pm \frac{b}{a}\left(x-h\right)+k$

This is the equation of a pair of straight lines; having slope $\pm \frac{b}{a}$ .

So, the graph of the given equation is a pair of straight lines; having slope $\pm \frac{b}{a}$ .

Now, as discussed, a hyperbola is formed when a plane intersects a double-sided cone in both nappes.

This is true for both a full-fledged hyperbola and the above equation.

The difference lies in the fact that the graph of the above equation is formed when the plane passes through the vertex of the cone.

Conclusion:

It has been explained how the given graph is similar to and different to a full-fledged hyperbola.