Question

Let the perpendiculars from any point on the line $2x+11y=5$ upon the lines $24x+7y-20=0$ and $4x-3y-2=0$ have the lengths $p_1$ and $p_2$ respectively. Then,

• A. $2p_1=p_2$
• B. $p_1=p_2$
• C. $p_1=2p_2$
• D. None of these

Answer 1

The correct option is B $p_1=p_2$

Let $P(x_1,y_1)$ be a point on the line $2x+11y=5$
$\Rightarrow 2x_1+11y_1=5$
$\Rightarrow x_1=\frac{5-11y_1}{2}....\left(1\right)$
Now , $p_1=\frac{|24x_1+7y_1-20|}{25}$
$\Rightarrow p_1=\frac{|40-125y_1|}{25}$....... (using $\left(1\right)$)
$\Rightarrow p_1=\frac{|8-25y_1|}{5}$
Now,$p_2=\frac{|4x_1-3y_1-2|}{5}$
$\Rightarrow p_2=\frac{|8-25y_1|}{5}$......(using $\left(1\right)$)

$\Rightarrow p_1=p_2$