# For points X (1,3,5) and (2,-4,6) find:

a) the coordinates of the point P that divides the segment XY (internally) in the ratio 3:2.

b) the coordinates of the point R that divides the segment XY (externally) in the ratio -2:5.

## To find the coordinates of the point P that divides the segment XY internally in the ratio 3:2, we need to use the concept of section formula in coordinate geometry.

a) Internal Division (3:2):

Let the coordinates of point P be (x,y,z).

According to the section formula, the coordinates of a point that divides a line segment with endpoints (x1, y1, z1) and (x2, y2, z2) internally in the ratio m:n is given by:

x = (m*x2 + n*x1)/(m + n)

y = (m*y2 + n*y1)/(m + n)

z = (m*z2 + n*z1)/(m + n)

In this case, the coordinates of point X are (1, 3, 5) and the coordinates of point Y are (2, -4, 6). The ratio is 3:2 (m = 3, n = 2).

Substituting the values into the formula, we get:

x = (3*2 + 2*1)/(3 + 2)

= (6 + 2)/(5)

= 8/5

y = (3*(-4) + 2*3)/(3 + 2)

= (-12 + 6)/(5)

= -6/5

z = (3*6 + 2*5)/(3 + 2)

= (18 + 10)/(5)

= 28/5

Therefore, the coordinates of point P are (8/5, -6/5, 28/5).

b) External Division (-2:5):

Similarly, to find the coordinates of the point R that divides the segment XY externally in the ratio -2:5, we use the section formula.

Let the coordinates of point R be (x,y,z).

According to the section formula, the coordinates of a point that divides a line segment with endpoints (x1, y1, z1) and (x2, y2, z2) externally in the ratio m:n is given by:

x = (m*x2 - n*x1)/(m - n)

y = (m*y2 - n*y1)/(m - n)

z = (m*z2 - n*z1)/(m - n)

In this case, the coordinates of point X are (1, 3, 5) and the coordinates of point Y are (2, -4, 6). The ratio is -2:5 (m = -2, n = 5).

Substituting the values into the formula, we get:

x = (-2*2 - 5*1)/(-2 - 5)

= (-4 - 5)/(-7)

= 9/7

y = (-2*(-4) - 5*3)/(-2 - 5)

= (8 - 15)/(-7)

= -7/7

= -1

z = (-2*6 - 5*5)/(-2 - 5)

= (-12 - 25)/(-7)

= -37/7

Therefore, the coordinates of point R are (9/7, -1, -37/7).