Consider the family of circles

$$x ^ { 2 } + y ^ { 2 } = 2 c x.$$

Show that the differential equation for determining the family of orthogonal trajectories is

$$\frac { d y } { d x } = \frac { 2 x y } { x ^ { 2 } - y ^ { 2 } }.$$

Find the differential equation of the given family. Find the orthogonal trajectories of the family. Use a graphing utility to graph both families on the same set of coordinate axes.

$$y = c _ { 1 } x$$

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. The family of orthogonal trajectories for a family of parallel lines is another family of parallel lines.

Draw a family tree of your immediate family. Next to each person, write a word to describe him or her. Then, write three sentences about your favorite relatives.

The Li family has four children born in different years. Find the probability that a family with four children has exactly two boys and two girls in any order.

Find the orthogonal trajectories of each given family of curves. In each case sketch several members of the family and several of the orthogonal trajectories on the same set of axes.

$$y^{2}=c x$$

Suppose that a family has 5 children. Also, suppose that the probability of having a girl is 1/2. Find the probabilities that the family has the following children. At least 3 boys

What role were woman expected to play in society and family life in the latter half of the nineteenth century, and how closely did patterns of family life correspond to this ideal?

A family buys airline tickets online. Each ticket costs $167. The family buys travel insurance with each ticket that costs$19 per ticket. The Web site charges a fee of $16 for the entire purchase. The family is charged a total of$1132. How many tickets did the famiiy buy?

Imagine that there is no full faith and credit clause and that your family intends to move to another state. In a short essay, explain how the another your family in your new residence.

A photographer lines up the 15 members of a family in a single line in order to take a photograph. How many different ways can the photographer arrange the family members for the picture?

Several of her family members died of tuberculosis.

A young couple wants to have a family of six children. Assuming that having a boy or girl is equally likely, what is the probability of having the following number of girls in their family?

$$2$$

The Chase family spends 20% of their income on rent and $\frac { 1 } { 3 }$ of their income on groceries. What percent of their income does the Chase family spend on rent and groceries combined?