Page 4 - LESSON NOTE

5. Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).

Ans:- Let P (x, y, z) be the point that is equidistant from points A(1, 2, 3) and B(3, 2, –1).

Now, PA = PB

2
2
=> PA = PB
2
2
2
2
2
2
=> (x - 1) + (y - 2) + (z - 3) = (x - 3) + (y - 2) + (z + 1)
2
2
2
2
2
2
=> x – 2x + 1 + y – 4y + 4 + z – 6z + 9 = x – 6x + 9 + y – 4y + 4 + z + 2z + 1
=> –2x –4y – 6z + 14 = –6x – 4y + 2z + 14
=> – 2x – 6z + 6x – 2z = 0
=> 4x – 8z = 0
=> x – 2z = 0
Thus, the required equation is x – 2z = 0
6. Find the coordinates of the point which divides the line segment joining the points (– 2, 3, 5) and (1,
– 4, 6) in the ratio (i) 2 : 3 internally, (ii) 2 : 3 externally.
Ans: (i) The coordinates of point R that divides the line segment joining points P(x1, y1, z1) andQ (x2, y2, z2)
internally in the ratio m : n are{(mx2 + nx1)/(m + n), (my2 + ny1)/(m + n), (mz2 + nz1)/(m + n)}
Let R (x, y, z) be the point that divides the line segment joining points (–2, 3, 5) and (1, –4, 6)
internally in the ratio 2 : 3

x = {2 * 1 + 3 * (-2)}/(2 + 3) = (2 - 6)/5 = -4/5

y = {2 * (-4) + 3 * 3}/(2 + 3) = (-8 + 9)/5 = 1/5

z = (2 * 6 + 3 * 5)/(2 + 3) = (12 + 15)/5 = 27/5

Thus, the coordinates of the required point are (-4/5, 1/5, 27/5).

(ii) The coordinates of point R that divides the line segment joining points P(x1, y1, z1) and
Q (x2, y2, z2) externally in the ratio m : n are{(mx2 - nx1)/(m - n), (my2 - ny1)/(m - n), (mz2 - nz1)/(m - n)}
Let R (x, y, z) be the point that divides the line segment joining points (–2, 3, 5) and (1, –4, 6)

externally in the ratio 2 : 3

x = {2 * 1 - 3 * (-2)}/(2 - 3) = (2 + 6)/(-1) = -8

y = {2 * (-4) - 3 * 3}/(2 - 3) = (-8 - 9)/(-1) = 17

z = (2 * 6 - 3 * 5)/(2 - 3) = (12 - 15)/(-1) = 3

Thus, the coordinates of the required point is (–8, 17, 3)