A line passes through the origin and makes equal angles with the positive coordinate axes.

Question

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $\mathrm{L}_1: 2 x+y+6=0$ and $\mathrm{L}_2: 4 x+2 y-p=0, p>0$

Accepted Answer

$\begin{gathered} 	an 	heta =|\frac{-2-1}{1+(-2)(1)}|=\frac{3}{1} \ \sin 	heta =\frac{3}{\sqrt{10}} \ AM=\frac{3}{\sqrt{10}}	imes \frac{9}{\sqrt{2}}=\frac{27}{2\sqrt{5}} \end{gathered}$

Dist. between ${I}_{1}\&{l}_{2},\, AM=|\frac{\frac{P}{2}+6}{\sqrt{5}}|=\frac{27}{2\sqrt{5}}$

$\begin{gathered} \frac{P}{2}+6=\pm \frac{27}{2} \ \frac{P}{2}=\frac{27}{2}-6\Rightarrow P=15,P>0 \ BM=\sqrt{\frac{81}{2}-{(\frac{27}{2\sqrt{5}})}^{2}} \ =\sqrt{\frac{810-729}{20}}=\sqrt{\frac{81}{20}}=\frac{9}{2\sqrt{5}} \ \frac{AM}{BM}=\frac{\frac{27}{\frac{2\sqrt{5}}{9}}}{\frac{2\sqrt{5}}{2\sqrt{2}}}=3 \end{gathered}$

Solutions · 1