KVPY Sample Paper KVPY Stream-SX Model Paper-7

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

A) with slope \[\frac{3}{2}\]

B) parallel to x-axis

C) with slope \[\frac{2}{3}\]

D) parallel to y-axis

Correct Answer: C

Solution :

Let point P is \[(a,\beta )\]and centroid of \[\Delta PQR\] is (h, k), then \[3h=\alpha +1+3\]and \[3k=\beta +4-2\]

\[\Rightarrow \] \[\alpha =3h-4\]and \[\beta =3k-2\]

Because \[(\alpha ,\beta )\]lies on \[2x-3y+4=0\]

\[\Rightarrow \]\[2(3h-4)-3(k-2)+4=0\]

\[\Rightarrow \]locus is\[6x-9y+2=0\] whose slope is\[\frac{2}{3}\].