# Locus

• March 20th 2010, 04:35 AM
deltaxray
Locus
P(x,y) is a variable point on the line 2x-3y+6 = o, and the point Q divides OP in the ratio 3:2. Show that Q has co ordinates ( \frac{3}{5}, \frac{2/5) (x+3)) hence find the locus of Q.

Thanks
• March 20th 2010, 07:13 AM
Hello deltaxray
Quote:

Originally Posted by deltaxray
P(x,y) is a variable point on the line 2x-3y+6 = o, and the point Q divides OP in the ratio 3:2. Show that Q has co ordinates ( \frac{3}{5}, \frac{2/5) (x+3)) hence find the locus of Q.

Thanks

I think you mean that Q has coordinates:
$\left( \tfrac{3}{5}x, \tfrac{2}{5} (x+3)\right)$
By similar triangles (or using the formula that you'll find in this post), the point Q that divides OP, where P is $(x,y)$, in the ratio $3:2$ is
$\left( \tfrac{3}{5}x, \tfrac{3}{5}y\right)$
But if $(x,y)$ lies on the line $2x-3y+6=0$, then
$y = \tfrac23(x+3)$
So Q is:
$\left( \tfrac{3}{5}x, \tfrac{3}{5}\cdot\tfrac23(x+3)\right)$
i.e.
$\left( \tfrac{3}{5}x, \tfrac{2}{5} (x+3)\right)$
So, if the coordinates of Q are $(h,k)$, then:
$h = \tfrac{3}{5}x$ and $k = \tfrac{2}{5} (x+3)$

$\Rightarrow x = \tfrac53h$

$\Rightarrow k = \tfrac25(\tfrac53h+3)$
$=\tfrac23h+\tfrac65$
$\Rightarrow 15k = 10h + 18$
So the equation of the locus of Q is:
$15y = 10x +18$