Question

Let $L$ be the line passing through the point $P(1,2)$ such that its intercepted segment between the co-ordinate axes is bisected at $P.$ If $L_1$ is the line perpendicular to $L$ and passing through the point $(-2,1),$ then the point of intersection of $L$ and $L_1$ is :

• A. $(\frac{11}{20},\frac{29}{10})$
• B. $(\frac{3}{10},\frac{17}{5})$
• C. $(\frac{3}{5},\frac{23}{10})$
• D. $(\frac{4}{5},\frac{12}{5})$

Answer 1

The correct option is D $(\frac{4}{5},\frac{12}{5})$

Let the equation of line $L$ be
$\frac{x}{a}+\frac{y}{b}=1$
Line cuts x-axis at $A(a,0)$ and $(0,b)$
Since, the intercepted segment between the axes is bisected at $P$.
So, coordinates of $P$ are $(\frac{a}{2},\frac{b}{2})$
Given coordinates of $P\equiv (1,2)$
$\Rightarrow \frac{a}{2}=1,\frac{b}{2}=2$
$\Rightarrow a=2,b=4$
Equation of line $L$ is
$\frac{x}{2}+\frac{y}{4}=1$
$\Rightarrow 2x+y=4$ .....(1)
Now, slope of $L_1$ is $\frac{1}{2}$
Equation of $L_1$ passing through $(-2,1)$ is
$y-1=\frac{1}{2}(x+2)$
$\Rightarrow 2y=x+4$ ....(2)
Solving (1) and (2), we get
$x=\frac{4}{5},y=\frac{12}{5}$
Point of intersection of $L$ and $L_1$ is $(\frac{4}{5},\frac{12}{5})$