Question

Find the coordinates of the point in which the line $2y-3x+7=0$ meets the line joining the two points $(6,-2)$ and $(-8,7)$. Find also the angle between them.

Answer 1

Equation of line joining $(6,-2)$ and $(-8,7)$

$y-(-2)=\frac{7-(-2)}{-8-6}(x-6)y+2=-\frac{9}{14}(x-6)14y+28=-9x+5414y+9x=\mathrm{26.....}\left(i\right)$

Slope of line $=m_1=-\frac{a}{b}=-\frac{9}{14}$

$2y-3x+7=0$

Slope of line $=m_2=-\frac{a}{b}=-\frac{-3}{2}=\frac{3}{2}$

$3x=2y+7$

Substituting in $\left(i\right)$

$14y+3(2y+7)=2614y+6y+21=2620y=5\Rightarrow y=\frac{1}{4}3x=2\left(\frac{1}{4}\right)+73x=\frac{15}{2}\Rightarrow x=\frac{5}{2}$

So the point of intersection is $(\frac{5}{2},\frac{1}{4})$

Angle between two lines i.e. $\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

$\tan \theta =\left|\frac{-\frac{9}{14}-\frac{3}{2}}{1+(-\frac{9}{14})\left(\frac{3}{2}\right)}\right|=\left|\frac{\frac{-18-42}{28}}{\frac{28-27}{28}}\right|=\left|\frac{\frac{-60}{28}}{\frac{1}{28}}\right|\tan \theta =60\Rightarrow \theta ={\tan }^{-1}60$