Hello deltaxray 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:$\displaystyle 15y = 10x +18$
Grandad