and PB and taking their ratios provided you know that A, P and B are collinear. Example : Find the coordinates of the points of trisection (i.e., points dividing in three equal parts) of the line segment joining the points A( , – ) and B(– , ). Solution : Let P and Q be the points of trisection of AB i.e., AP = PQ = QB (see Fig. .
). Therefore, P divides AB internally in the ratio : . Therefore, the coordinates of P, by applying the section formula, are ( ) ( ) ( ) ( ) , i.e., (– , ) Now, Q also divides AB internally in the ratio : . So, the coordinates of Q are ( ) ( ) ( ) ( ) , i.e., (– , ) Fig.
. Therefore, the coordinates of the points of trisection of the line segment joining A and B are (– , ) and (– , ). Note : We could also have obtained Q by noting that it is the mid-point of PB. So, we could have obtained its coordinates using the mid-point formula.
Example : Find the ratio in which the y -axis divides the line segment joining the points ( , – ) and (– , – ). Also find the point of intersection. Solution : Let the ratio be k : . Then by the section formula, the coordinates of the point which divides AB in the ratio k : are This point lies on the y -axis, and we know that on the y -axis the abscissa is .
Therefore, = k = That is, the ratio is : . Putting the value of k = , we get the point of intersection as , . Example : If the points A( , ), B( , ), C( , ) and D( p , ) are the vertices of a parallelogram, taken in order, find the value of p . Solution : We know that diagonals of a parallelogram bisect each other.
So, the coordinates of the mid-point of AC