Question

A straight line L through the origin meets the lines `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 `2x-y- 5` and `3x +y 5` respectively. Lines `L_1` and `L_2` intersect at R. Locus of R, as L varies, is

Answer 1

Let the equation of straight line `L` be
`y=mx`
`P=((1)/(m+1),(m)/(m+1))`
`Q=((3)/(m+1),(3m)/(m+1))`

Now, equation of
`L_(1) : y-2x=(m-2)/(m+1)`……`(i)`
and equation of
`L_(2) : y+3x=(3m+9)/(m+1)`....`(ii)`
By eliminating `m` from Eqs. `(i)` and `(ii)`, we get locus of `R` as `x-3y+5=0`, which represents a straight line.