 Hi and welcome to the session let's work out the following question. The question says the line segment joining the points 3 minus 4 and 1 2 is trisected at the points p and q. If the coordinates of p and q are p 2 and 5 by 3 q respectively find the values of p and q. Let us start with a solution to this question. So let the coordinate of point a be 3 minus 4 b b 1 2 p is given to be small p 2 and q 5 by 3 q. So this is the line we have a p q and b are the points on this line. Since p and q are given to be trisecting a b therefore we have a p is equal to p q is equal to q b is equal to 1 by 3 a b or we have a p is equal to 1 by 3 a b this we call 1 and we have b q is equal to 1 by 3 a b we call this 2 but by the distance formula we say that a b is the distance between the point a and b and that is equal to square root of 1 minus 3 the whole square plus 2 minus minus 4 the whole square that is equal to square root of minus 2 the whole square plus 2 plus 4 the whole square that is equal to square root of now square of minus 2 is 4 plus square of 6 is 36 that is equal to square root of 40. Now a p that is distance between the point a and p will be equal to square root of p minus 3 the whole square plus minus 2 minus minus 4 the whole square that is equal to square root of p minus 3 the whole square plus minus 2 plus 4 the whole square that is equal to square root of p minus 3 the whole square plus 2 square that is equal to square root of p minus 3 the whole square plus 4 or we can say that AP is equal to square root of p minus 3 the whole square plus 4 and similarly VQ is equal to square root of 5 by 3 minus 1 the whole square plus Q minus 2 the whole square that is equal to square root of 5 minus 3 divided by 3 the whole square plus Q minus 2 the whole square this is equal to square root of 2 by 3 the whole square plus Q minus 2 the whole square and that is equal to square root of 4 by 9 plus Q minus 2 the whole square now putting the values of AB, AP and QB in 1 and 2 that is this equation and this equation we get square root of p minus 3 the whole square plus 4 is equal to 1 by 3 into square root of 40 on square in both the sides we get p minus 3 the whole square plus 4 is equal to 1 by 9 into 40 or we can say that p minus 3 the whole square is equal to 40 divided by 9 minus 4 now this implies that p minus 3 the whole square is equal to 40 minus 36 divided by 9 this further implies that p minus 3 the whole square is equal to 4 divided by 9 or we can say that p minus 3 is equal to plus minus square root of 4 by 9 that is equal to plus minus 2 by 3 now we have either p is equal to 2 by 3 plus 3 or p is equal to minus 2 by 3 plus 3 now this will imply that p is equal to 2 plus 9 divided by 3 and this implies that p is equal to minus 2 plus 9 the whole divided by 3 now from this one we have p is equal to 11 by 3 and from here we have p is equal to 7 by 3 therefore p is equal to 11 by 3 or 7 by 3 also we have square root of 4 by 9 plus q minus 2 the whole square is equal to 1 by 3 into square root of 40 now squaring both the sides we get 4 by 9 plus q minus 2 the whole square is equal to 40 divided by 9 now this further implies that q minus 2 the whole square is equal to 40 by 9 minus 4 by 9 that is equal to 36 divided by 9 now taking square root on both the sides we get q minus 2 is equal to plus minus square root of 36 by 9 that is equal to plus minus now 36 is the square of 6 and 9 is square of 3 so we have plus minus 6 by 3 and this further simplifies to plus minus 2 now we have q is equal to 2 plus 2 or q is equal to minus 2 plus 2 now in this case we have q is equal to 4 or we have q is equal to 0 therefore our answer to this question is p is equal to 11 by 3 or 7 by 3 and q is equal to 4 or 0 so this is our answer to this question I hope that you understood the solution and enjoyed the session have a good day.