Question

A ine through H(3, 1) and J(5, a) has positive slope and makes a 60° angle measured counterclockwise with the positive x-axis. Find the value of a.

Short answer

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

Step-by-step solution

Step-1 – Given

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

Step-2 – To determine

We have tofind the value ofa.

Step-3 – Calculation 

The slope formula with two points$(x_1,y_1)$and$(x_2,y_2)$is

$m=\frac{y_2-y_1}{x_2-x_1}$

The given points are$H(3,1)$and $J(5,a)$.

We know that, the slope of a line is defined as “the tangent of the angle which the line makes with positive direction of the$x-axis$in the anticlockwise direction”.

It means$m=\tan \theta$

So,

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

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