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# (i) In what ratio does the point (5, 4) divide the line segment joining the points (2, 1) and (7 ,6) ? (ii) In what ratio does the point ( – 4, b) divide the line segment joining the points P (2, – 2), Q ( – 14, 6) ? Hence find the value of b.

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M.L Aggarwal book Important Question of class 10 chapter Based on Section Formula for ICSE BOARD.
You have to find that in what ratio does a point divides the line segment joining the two points.
This is the Question Number 16, Exercise 11 of M.L Aggarwal.

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1. (i) Let the ratio that the point (5,4) divide the line segment joining the points (2,1) and (7,6) be m:n,

Here x1 = 2 , y1 = 1 , x2 = 7, y2 = 6, x = 5, y = 4

By section formula, x = (mx2+nx1)/(m+n)

5 = (m×7+n×2)/(m+n)

5 = (7m+2n)/(m+n)

5(m+n) = 7m+2n

5m+5n = 7m+2n

5m-7m = 2n-5n

-2m = -3n

m/n = -3/-2 = 3/2

Hence the ratio m:n is 3:2.

(ii) Let the ratio that the point (-4,b) divide the line segment joining the points (2,-2) and (-14,6) be m:n,

Here x1 = 2 , y1 = -2 , x2 = -14, y2 = 6, x = -4, y = b

By section formula, x = (mx2+nx1)/(m+n)

-4 = (m×-14+n×2)/(m+n)

-4= (-14m+2n)/(m+n)

-4(m+n) = -14m+2n

-4m-4n = -14m+2n

-4m+14m = 2n+4n

10m = 6n

m/n = 6/10 = 3/5

Hence the ratio m:n is 3:5.

By Section formula y = (my2+ny1)/(m+n)

b= (3×6+5×-2)/(3+5)

b = (18-10)/8

b = 8/8

b = 1

Hence the value of b is 1 and the ratio m:n is 3:5.

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