Question

Find the equation of the line passing through the point $(2,3)$ and making intercept of length $3$ unit between the lines $y+2x=2$ and $y+2x=5$.

Answer 1

Dist AB$=\frac{|c_1-c_2|}{\sqrt{1-m^2}}$

$=\frac{|5-2|}{\sqrt{1+2^2}}$

$=\frac{3}{\sqrt{5}}$

$AC=3$ given

$\therefore CB=\sqrt{A{C}^2-A{B}^2}$ [Pythagoras Theorem]

$=\frac{6}{\sqrt{5}}$

$\therefore \tan \theta =\frac{\frac{3}{\sqrt{5}}}{\frac{6}{\sqrt{5}}}=\frac{1}{2}$

$\therefore \theta ={\tan }^{-1}\frac{(m_1-m_3)}{1+m_1{m}_3}$ $[m_3\to s1]$

$\Rightarrow \tan \theta =\frac{2-m_3}{1+2m_3}$

$\Rightarrow \frac{1}{2}=\frac{2-m_3}{1+2m_3}$

$\Rightarrow m_3=1$

$\frac{y-3}{x-2}=m_3$

$\Rightarrow \frac{y-3}{x-2}=1$

$\Rightarrow y-x=1$.