The line segment joining the points A(3,-4) and B(1,2) is trisected at the points P(p, -2) and Q `(5/3,q).` . Find the values of p and q.

#### Solution

Let P and Q be the points of trisection of AB.

Then, P divides AB in the radio 1:2

So, the coordinates of P are

` x= ((mx_2 +nx_1))/((m+n)) , y = ((my_2+ny_1))/((m+n))`

` ⇒ x = ({ 1 xx 1+2xx(3)})/(1+2) , y = ({1 xx 2+2xx(-4)})/(1+2)`

` ⇒ x = (1+6)/3 , y (2-8)/3`

` ⇒ x = 7/3 , y -6/3`

` ⇒x =7/3 , y =-2`

Hence, the coordinates of P are `(7/3 , -2)`

But ,(p -2) are the coordinates of P.

so , p=`7/3`

Also, Q divides the line AB in the ratio 2:1

So, the coordinates of Q are

`x = ((mx_2 +mx_1))/((m+n)) , y = ((my_2+my_1))/((m+n))`

`⇒x = ((2xx1+1xx3))/((2+1)) , y = ({ 2xx2+1xx(-4)})/(2+1)`

`⇒ x = (2+3)/3 , y = (4-4)/3`

`⇒ x = 5/3 , y =0`

Hence, coordinates of Q are `(5/3, 0)`

But the given coordinates of Q are `(5/3,q)`

so , q=0

Thus , `p=7/3 and q =0`