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Question

(i) Points P and Q trisect the line segment joining the points A(–2, 0) and B(0, 8) such that P is near to A. Find the coordinates of P and Q.
(ii) The line segment joining the points (3, −4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, −2) and (5/3, q) respectively. Find the values of p and q.

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Solution

(ii) Let the points A(3, −4) and B(1, 2) is trisected at the points P(p, −2) and Q(5/3, q).

Thus, AP = PQ = QB

Therefore, P divides AB internally in the ratio 1 : 2.

Section formula: if the point (x, y) divides the line segment joining the points (x1, y1) and (x2, y2) internally in the ratio m : n, then the coordinates (x, y) = mx2+nx1m+n, my2+ny1m+n

Therefore, using section formula, the coordinates of P are:

p, -2=11+231+2,12+2-41+2p, -2=1+63,2-83p, -2=73,-63p, -2=73,-2p=73

Also, Q divides AB internally in the ratio 2 : 1.

Therefore, using section formula, the coordinates of Q are:

53, q=21+132+1,22+1-42+153, q=2+33,4-4353, q=53,0353, q=53,0q=0

Hence, the values of p and q is 73 and 0, respectively.

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