#### Answers

*x*

_{ 1 },

*y*

_{ 1 }) and B (

*x*

_{ 2 },

*y*

_{ 2 }) and assume that P (

*x*,

*y*) divides AB internally in the ratio

*m*:

*n*i.e. PA: PB =

*m*:

*n*

*x*-axis. Draw AQ and PC perpendiculars to PS and BT respectively.

Let

Unknown control sequence 'fracPRRQ ' .

Draw PM, QN and RL perpendiculars to the XY-plane. Through R, draw a straight line ARB parallel to MLN to meet MP (produced) and NQ in points A and B respectively.

[Since PM, QN, RL are perpendiculars to the XY-plane, are parallel,and as they are cut by the line PRQ, they are coplanar. Points M,L, N lie on a straight line which is the intersection of this plane with XY-plane. Therefore, a line through R parallel to MLN also lies in that plane and hence meets MP (produced) and NQ.]

Clearly,

Unknown control sequence 'fracPABQ ' ..............(i)

From figure,

Unknown control sequence 'fracz '

Unknown control sequence 'fracm ' .

Similarly, Unknown control sequence 'fracm ' and Unknown control sequence 'fracm ' .

Hence the co-ordinates of R are

Unknown control sequence 'fracm '

## Remark

If

## Rule to write down the co-ordinates of the point which divides the join of two given points in a given ratio:

Draw any line segment and write down the co-ordinates of the given points P and Q at its extremities. Let R(x,y,z) be the point which divides [PQ] in the ratio

For x co-ordinate of R, multiply

Thus, Unknown control sequence 'fracm ' .

Similarly, for y and z co-ordinates.

## Remark

For problems in which it is required to find out the ratio when a given point divides the join of two given points, it is convenient to take the ratio as