Question

Find the distance of the line $4x-y=0$ find the point $P(4,1)$ measured along the line making an angle of ${135}^o$ with the positive x-axis.

Answer 1

$P(4,1)$

$m=tan135=tan({160}^{\circ }-{45}^{\circ })=-tan{45}^{\circ }=-1$

equation of line with slope $\left(135\right)\&P(4,1)$

$\Rightarrow \frac{y-1}{x-4}=-1$

$\Rightarrow y-1=-x+4$

$\Rightarrow x+y-5=0\Rightarrow x+y=5$ ........(i)

Now,

$4x-y=0$

(i) + (ii)

$5x=5\Rightarrow x=1$ & $x+y=5$

$y=5-1=4$

So,

$P(4,1),Q(1,4)$

$Distance=\sqrt{(4-1{)}^2+(1-4{)}^2}=3\sqrt{2}units$