# Locus

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

$\displaystyle \Rightarrow x = \tfrac53h$

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