Question

A straight line is drawn through $p(3,4)$ to meet the axes of x and y at A and B respectively. If the rectangle OACB Is completed, then find the locus of C?

Answer 1

Given:

$P(3,4)\equiv (x,y)$

Let slope of line be $m$

$\Rightarrow$Equation of line is,

$\Rightarrow$$y-y_1=m(x-x_1)$

$\Rightarrow$$y-4=m(x-3)$

$atpointA,y=0$
$\therefore x=\left(\frac{-4}{m}\right)+3$
$atpointB,x=0$
$\therefore y=-3m+4$
$\therefore h=\left(\frac{-4}{m}\right)+3andK=-3m+4$
$orh-3=\left(\frac{-4}{m}\right)$
$orm=\left(\frac{-4}{h-3}\right)$
$andK=-3\left(\frac{-4}{h-3}\right)+4$
$K=\left(\frac{12+4h-12}{h-3}\right)$
$orK(h-3)=4h$
$ory(x-3)=4xrequiredlocus$