Question

Find the distance of the line 4x - y = 0 from the point p(4,1) measured along the line making an angle of 135 degree with the positive axis

Answer 1

AnswEr:

The equation in distance from of the line passing through p(4,1) and making and angle of 135° with the positive x-axis is

$\qquad \tt \frac{x - 4}{ \cos(135) \degree } = \frac{y - 1}{ \sin(135) \degree } = r$

Suppose it cuts 4x - y = 0 at Q such that PQ = r. Then, the coordinates of Q are given by

$\qquad \tt \frac{x - 4}{ \cos(135) \degree } = \frac{y - 1}{ \sin(135) \degree } = r$

$\implies \tt \frac{x - 4}{ - 1/ \sqrt{2} } = \frac{y - 1}{1/ \sqrt{2} } = r \\ \\ \tt \implies \: x = 4 - \frac{r}{ \sqrt{2} } \: and \: y = 1 + \frac{r}{ \sqrt{2} }$

So, the coordinates of Q are -

$\tt(4 - \frac{r}{ \sqrt{2} } \: and \: 1 + \frac{r}{ \sqrt{2} })$

Clearly, Q lies on 4x - y = 0.

$\therefore \tt \: 16 - \frac{4r}{ \sqrt{2} } - 1 - \frac{r}{ \sqrt{2} } = 0 \\ \\ \implies \tt \: \frac{5r}{ \sqrt{2} } = 15 \\ \\ \tt \implies r = 3 \sqrt{2}$

Hence, required distance is 3√2 units.