Intersection of two lines, each defined by a point and an angle

Question

How to find a point where $L_1$ and $L_2$ intersect, given that $L_1$ goes through $(x_1,y_1)$ and it's slope is $\alpha_1$ and $L_2$ goes through $(x_2,y_2)$ and it's slope is $\alpha_2$? I tried applying sine theorem but I get two answers instead of one. Doing to Cartesian coordinates complicates things (vertical and horizontal lines should be differently to avoid division by zero in $\dfrac{1}{\sin(\alpha)}$ and $\dfrac{1}{\cos(\alpha)}$ cases.

Answer 1

Initially, we have $L_1: y – y_1 = m_1(x – x_1)$ and $ L_2: y – y_2 = m_2(x – x_2)$.

If they intersect at (h, k), then just combine the two equations to get $ m_1(h – x_1) + y_1 = m_2(h – x_2) + y_2$

Finally, $h = \dfrac {(m_2x_2 – m_1x_1) – (y_2 – y_1)}{m_2 – m_1}$

$k$ can be found by substituting the value of $h$ back in $L_1$.

Edit:-

The following cases should be checked before applying the formula.

1) If $m_1 = 0$, then from $L_1$, $k = y_1$ and h can be found using $L_2$.

2) If $m_2 = 0$, then ....

3) If $m_1 = 0$ and $m_2 = 0$, then either they never meet or meet at infinitely many points.

4) If $m_1 = \infty$ (i.e. $L_1: x = x_1$), then simply $h = x_1$, and k can be found accordingly.

5) If $m_2 = \infty$, then .....

6) If $m_1 = \infty$ and $m_2 = \infty$, then ......

7) If $m_1 – m_2 = 0$, this means the two lines are either parallel or actually the same line. In the first occasion, the point of intersection can never be found. In the second occasion, there are infinitely many points of intersection.