ICSE Board Question Based on Section Formula Chapter of M.L Aggarwal for class10

In this question following parts to be solved

(i) Calculate the ratio in which the line segment joining two points is divided by the y-axis

(ii) In what ratio does a line divide the line segment joining the points two points ? Also, find the coordinates of the point of division.

This is the Question Number 19, Exercise 11 of M.L Aggarwal.

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# (i) Calculate the ratio in which the line segment joining (3, 4) and( – 2, 1) is divided by the y-axis. (ii) In what ratio does the line x – y – 2 = 0 divide the line segment joining the points (3, – 1) and (8, 9)? Also, find the coordinates of the point of division.

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(i) Let m:n be the ratio in which the line segment joining (3,4) and (-2,1) is divided by the Y axis.

Since the line meets Y axis, its x co-ordinate is zero.

Here x

_{1 }= 3, y_{1}= 4x

_{2}= -2, y_{2}= 1By section formula,x = (mx_{2}+nx_{1})/(m+n)0 = (m×-2+n×3)/(m+n)

0 = (-2m+3n)/( m+n)

0 = -2m+3n

2m = 3n

m/n = 3/2

Hence the ration m:n is 3:2.

(ii)Let the line x-y-2 = 0 divide the line

segment joining the points (3,-1) and (8,9) in the ratio m:n at the point P(x,y)Here x

_{1 }= 3, y_{1}= -1x

_{2}= 8 y_{2}= 9By section formula,x = (mx_{2}+nx_{1})/(m+n)x = (m×8+n×3)/(m+n)

x = (8m+3n)/( m+n) …. (i)

By Section formula y = (my

_{2}+ny_{1})/(m+n)y = (m×9+n×-1)/(m+n)

y = (9m-n)/(m+n) …. (ii)

Since the point P(x,y) lies on the line x-y-2 = 0,

eqn (i) and (ii) will satify the equation x-y-2 = 0 …(iii)

Substitute (i) and (ii) in (iii)

[(8m+3n)/( m+n)] – [(9m-n)/(m+n)]-2 = 0

[(8m+3n)/( m+n)] – [(9m-n)/(m+n)]-[2(m+n)/(m+n)] = 0

8m+3n-(9m-n)-2(m+n) = 0

8m+3n-9m+n-2m-2n = 0

-3m+2n = 0

-3m = -2n

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

Hence the ratio m:n is 2:3.

Substitute m and n in (i)x = (8m+3n)/( m+n)

x = (8×2+3×3)/(2+3)

x = (16+9)/5

x = 25/5 = 5

Substitute m and n in (ii)y = (9m-n)/(m+n)

y = (9×2-3)/(2+3)

y = (18-3)/5

y = 15/5 = 3

Hence the co-ordinates of P are (5,3).