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A point P moves on the line 2x-3y+4=0. I...

A point P moves on the line 2x-3y+4=0. If Q(1,4) and R (3,-2) are fixed points, then the locus of the centroid of `Delta PQR` is a line

A

with slope `(2)/(3)`

B

with slope `(3)/(2)`

C

parallel to `Y`-axis

D

parallel to `X`-axis

Text Solution

Verified by Experts

Let the coordinates of point `P` be `(x_(1),y_(1))`
`:'P` lies on the line `2x-3y+4=0`
`:.2x_(1)-3y_(1)+4=0`
`impliesy_(1)=(2x_(1)+4)/(3)`………`(i)`
Now, let the centroid of `DeltaPQR` be `G(h,k)`, then
`h=(x_(1)+1+3)/(3)`
`impliesx_(1)=3h-4` ........`(ii)`
and `k=(y_(1)+4-2)/(3)`
`impliesk=((2x_(1)+4)/(3)+2)/(3)` [from Eqs.`(i)`]
`implies3k=(2x_(1)+4+6)/(3)`
`implies9k-10=2x_(1)` .......`(iii)`
Now, from Eqs. `(ii)` and `(iii)`, we get
`2(3h-4)=9k-10`
`implies6h-8=9k-10`
`implies6h-9h+2=0`
Now, replace `h` by `x` and `k` by `y`.
`implies6x-9y+2=0`. which is the required locus and slope of this line is `(2)/(3)` [`:' "slope of"ax+by+c=0"is"-(a)/(b)`]
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