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PROVE SECTION FORMULA. 

Here is the proof of the section formula.
 
Consider any two points A (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
Draw AR, PS and BT perpendicular to the x-axis. Draw AQ and PC perpendiculars to PS and BT respectively.
 
 
In ∆PAQ and ∆BPC
∠PAQ = ∠BPC    (pair of corresponding angles)
∠PQA = ∠BCP    (90 °)
Hence, ∆PAQ ∼ ∆BPC  (AA similarity criterion)
 
Hope! You got the concept.
Best Wishes @!
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View Full Answer

 Let P(x1y1z1) and Q(x2y2z2) be the two given points. Let R(x,y,z) divide [PQ] in the given ratio m1:m2 such that

Unknown control sequence 'fracPRRQ'.

 

SECTION FORMULA

 

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, s APR and RBQ are similar,

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

From figure,

 PA=MA−MP=LR−MP=z−z1 and

 BQ=NQ−NB=NQ−LR=z2−z.

  From (i), we get

 Unknown control sequence 'fracz'

(m1+m2)z=m1z2+m2z

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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 m1:m2is positive, then segments PR and RQ have same direction and hence R divides [PQ] internally [shown in fig 1.5(i)]

 

SECTION FORMULA

If m1:m2 is negative, then segments PR and RQ have opposite directions and hence R divides [PQ] externally [shown in fig 1.5(ii)]

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 m1:m2.

 

SECTION FORMULA

For x co-ordinate of R, multiply m1with x2and m2 with x1 as shown in fig 1.6 by arrow heads and add the products. Divide the sum by m1+m2.

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 k:1(k=−1), for, in this way to two unknown (m1and m2) are reduced to one and the co-ordinates of R become

k+1kx2+x1k+1ky2+y1k+1

  • 2
i think Afzal is write answer. 
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i am satisfied  Prakhar answer .
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