KVPY Sample Paper KVPY Stream-SX Model Paper-3

Consider the set of all lines \[\text{p}x+qy+r=0\] such that \[3p+2q+4r=0.\] Which one of the following statements is true?

A) The lines are all parallel

B) Each line passes through the origin.

C) The lines are concurrent at the point

D) \[\left( \frac{3}{4},\frac{1}{2} \right)\]

Correct Answer: D

Solution :

\[4Px\,+\,4qy\,+\,4r\,=\,0\]and \[4r\,=\,-3p-\,2q\]

\[\Rightarrow \] \[P\,(4x-3)\,+q\,(4y-2)\,=\,0\]

This is equation of family of lines passing through

point of intersection of the lines'

\[4x\,-\,3\,=\,0\,and\,4y\,-\,2\,=\,0\]

i.e., \[\left( \frac{3}{4},\frac{1}{2} \right).\]