Question

A line $L_1$ passes through the points $(1,1)$ and $(2,0)$ and another line $L_2$ passes through $(\frac{1}{2},0)$ and is perpendicular to $L_1.$ If the area of the triangle formed by the line $L_1,L_2$ and $y-$axis is $\frac{p}{q}$ sq. units where $p,q$ are co-prime, then value of $p+q$ is

Answer 1

Line $L_1$ which passes through $(1,1)$ and $(2,0)$ is
$y-1=\frac{-1}{1}(x-1)$
$\Rightarrow y-1=-x+1$
$\Rightarrow x+y=2\cdots \left(1\right)$

Line $L_2$ is $y-0=1(x-\frac{1}{2})$
$\Rightarrow 2y=2x-1$
$\Rightarrow 2x-2y=1\cdots \left(2\right)$
On solving equation $\left(1\right)$ and $\left(2\right),$
$x=\frac{5}{4},y=\frac{3}{4}$


Required area $=\frac{1}{2}\times$ base $\times$ height
$=\frac{1}{2}\times \frac{5}{2}\times \frac{5}{4}=\frac{25}{16}$ sq. units