Chapter 10.6, Problem 1PT

The figure at the right shows one point P on a conic and the distances of P to the focus and the directrix of a conic. What type of conic is it?

1. a) parabola
2. b) ellipse
3. c) hyperbola
4. d) not enough information is provided to answer the question.

To determine

To identify: The correct option for the blank in the statement, “The type of conic shown in the figure is a ____”.

Answer to Problem 1PT

The type of conic shown in the figure is a hyperbola which is option c.

Explanation of Solution

Given:

The options are, a) parabola, b) ellipse, c) hyperbola and d) not enough information is provided to answer the question.

Formula used:

Let PF be the distance between the point P on a conic and the focus F and, let Pl be the distance between the point on a conic and the directrix l. Then,

The conic is a parabola if the ratio $\frac{\left|PF\right|}{\left|Pl\right|}=1$.

The conic is an ellipse if the ratio $\frac{\left|PF\right|}{\left|Pl\right|}=e<1$.

The conic is a hyperbola if the ratio $\frac{\left|PF\right|}{\left|Pl\right|}=e>1$.

Calculation:

From the given figure, it is observed that the distance between the point P on a conic and the focus, F is 6 and the distance between the point on a conic and the directrix l is 4.

Compute the value of the ratio $\frac{\left|PF\right|}{\left|Pl\right|}$ as follows.

$\begin{array}{c}\frac{\left|PF\right|}{\left|Pl\right|}=\frac{6}{4}\\ =\frac{3}{2}\\ =1.5\\ >1\end{array}$

Therefore, the type of conic shown in the figure is a hyperbola which is option c.