Straight Line And Pair Of Straight Lines Ques 48

  1. A straight line $L$ through the origin meets the line $x+y=1$ and $x+y=3$ at $P$ and $Q$ respectively. Through $P$ and $Q$ two straight lines $L _1$ and $L _2$ are drawn, parallel to $2 x-y=5$ and $3 x+y=5$, respectively. Lines $L _1$ and $L _2$ intersect at $R$, show that the locus of $R$ as $L$ varies, is a straight line.

$(2002,5$ M)

Show Answer

Solution:

Formula:

Equation Of A Straight Line In Various Forms:

  1. Let the equation of straight line $L$ be

$Q \equiv \frac{3}{m+1}, \frac{3 m}{m+1}$

Now, equation of

$$ L _1: y-2 x=\frac{m-2}{m+1} $$

and equation of

$$ L _2: y+3 x=\frac{3 m+9}{m+1} $$

By eliminating $m$ from Eqs. (i) and (ii), we get locus of $R$ as $x-3 y+5=0$, which represents a straight line.

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