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.

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- Mar 20th 2010, 04:35 AMdeltaxrayLocus
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 AMGrandad
Hello deltaxrayI 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 $\displaystyle k = \tfrac{2}{5} (x+3)$So the equation of the locus of Q is:

$\displaystyle \Rightarrow x = \tfrac53h$

$\displaystyle \Rightarrow k = \tfrac25(\tfrac53h+3)$$\displaystyle =\tfrac23h+\tfrac65$$\displaystyle \Rightarrow 15k = 10h + 18$$\displaystyle 15y = 10x +18$Grandad