`Q` is a variable point whose locus is `2x+3y+4=0;` corresponding to a particular position of `Q ,P` is the point of section of `O Q ,O` being the origin, such that `O P: P Q=3: 1.` Find the locus of `Pdot`

Answer

`2x+3y+3=0`

Explanation

Let Q be the point `(X,Y)` and P be the point (x,y), the coordinates of Q satisfy the equation `2x+3y+4=0`, so that `2X+3Y+4=0`.
Applying the section formula for OQ,O being (0,0) we get
`x=(0+3X)/(1+3),y=(0+3Y)/(1+3)`
from which we get
`X=(4)/(3)x,Y=(4)/(3)y`
Substituting these values, the locus of P is
`(8)/(3)x+4y+4=0`
or `2x+3y+3=0`