Chapter 13.2, Problem 33WE

To determine

To find the value of $a$ .

Answer to Problem 33WE

The value of $a$ is $2\sqrt{3}+1$

Explanation of Solution

Given:

A line through points $H\left(3,1\right)$ and $J\left(5,a\right)$ has positive slope and makes a ${60}^{°}$ angle measured counterclockwise with the positive $x-axis$ .

Calculation:

Slope of a line when two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are given is

  $m=\frac{y_2-y_1}{x_2-x_1}$

Points are $H\left(3,1\right)$ and $J\left(5,a\right)$

Also, Slope of line is defined as a the tangent of angle which the line makes with positive direction of $x-axis$ in anticlockwise direction.

  $m=\tan \theta$

So,

  $\begin{array}{l}\tan \theta =\frac{y_2-y_1}{x_2-x_1}\\ \tan {60}^{\circ }=\frac{a-1}{5-3}\\ \sqrt{3}=\frac{a-1}{2}\\ a=2\sqrt{3}+1\end{array}$

Therefore, the value of $a$ is $2\sqrt{3}+1$ .