Question

A variable line through the point (6/5, 6/5) the co-ordinate axes at the poiunt A and B. If the point P divides AB intermally in the ratio 2 : 1, show that the locus of p is $5xy=2(2x+y)$

Answer 1

Let lines slope be m

It equation $y-\frac{6}{5}=m(x-\frac{6}{5})$

$A:0-\frac{6}{5}=mx-\frac{6m}{5}$

$\Rightarrow x=\frac{6(m-1)}{5m}$

$B:y=\frac{6}{5}-\frac{6m}{5}=\frac{6}{5}(1-m)$

$h=\frac{x}{3}=\frac{6(m-1)}{15m}$

$R=\frac{2y}{3}=\frac{2}{3}\times \frac{6}{5}(1-m)$

$=\frac{4(1-m)}{5}$

$\Rightarrow 5k=4-4m$

$\Rightarrow m=\frac{4-5k}{4}$

$\Rightarrow h=\frac{2}{5}(1-\frac{4}{4-5k})$

$\Rightarrow h=\frac{2}{5}\left(\frac{-5k}{4-5k}\right)$

$\Rightarrow (20-25k)h=-10k$

$\Rightarrow (4-5k)x=-2y$

$\Rightarrow 5xy=4x+2y$.