If θ is the angle between the lines joining the points
Question:

If θ is the angle between the lines joining the points (0, 0) and B(2, 3), and the points C(2, -2) and D(3, 5), show that

$\tan \theta=\frac{11}{23}$

Solution:

The given points are A(0,0),B(2,3) and C(2,-2),D(3,5).

slope $=\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)$

The slope of line $\mathrm{AB}$ is $\left(\frac{3-0}{2-0}\right)=\frac{3}{2}=\mathrm{m}_{1}$

And the slope of line $C D$ is $\left(\frac{5+2}{3-2}\right)=7=\mathrm{m}_{2}$

We know that angle between two lines with their slopes as m1 and m2 is given by

$\tan \theta=\frac{\mathrm{m}_{2}-\mathrm{m}_{1}}{1+\mathrm{m}_{1} \mathrm{~m}_{2}}$

$=\frac{7-\frac{3}{2}}{1+7 \times \frac{3}{2}}$

$=\frac{\frac{14-3}{2}}{\frac{2+21}{2}}$

$=\frac{11}{23}$

$\Rightarrow \tan \theta=\frac{11}{23}$

Hence proved