4. Find the ratio in which the point P(x,2) divides the line segment joining the points B(4,-3) and A(12,5)? Also find the value of x
1. Find the ratio in which the point `P(x,2)` divides the line segment joining the points `A(12,5)` and `B(4,-3)`? Also find the value of x.
Solution:
Method-1 : considering the ratio `m:n`
Let the point `P(x,2)` divides the line segment joining the points `A(12,5)` and `B(4,-3)` in the ratio `m:n`
The given points are `A(12,5),B(4,-3)`
`:. x_1=12,y_1=5,x_2=4,y_2=-3`
Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`
`:.P(x,2)=((m(4)+n(12))/(m+n),(m(-3)+n(5))/(m+n))`
`:. (m(-3)+n(5))/(m+n)=2`
`:. -3m+5n=2(m+n)`
`:. -3m+5n=2m+2n`
`:. -3m-2m=2n-5n`
`:. -5m=-3n`
`:. 5m=3n`
`:. m/n=(3)/(5)`
:. The point divides the line joining `A(12,5)` and `B(4,-3)` in the ratio `3:5`
Putting `m=3,n=5`, we get the coordinates of point P
`x=(mx_2+nx_1)/(m+n)`
`=(3*4+5*12)/(3+5)`
`=(12+60)/(8)`
`=(72)/(8)`
`=9`
Thus, the value of x is `9`
Method-2 : considering the ratio `k:1`
Let the point `P(x,2)` divides the line segment joining the points `A(12,5)` and `B(4,-3)` in the ratio `k:1`
The given points are `A(12,5),B(4,-3)`
`:. x_1=12,y_1=5,x_2=4,y_2=-3`
Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`
`:.P(x,2)=((k(4)+(12))/(k+1),(k(-3)+(5))/(k+1))`
`:. (k(-3)+(5))/(k+1)=2`
`:. -3k+5=2(k+1)`
`:. -3k+5=2k+2`
`:. -3k-2k=2-5`
`:. -5k=-3`
`:. 5k=3`
`:. k=(3)/(5)`
:. The point divides the line joining `A(12,5)` and `B(4,-3)` in the ratio `3:5`
Putting `k=3/5`, we get the coordinates of point P
`x=(kx_2+x_1)/(k+1)`
`=(3/5*4+12)/(3/5+1)`
`=(3*4+12*5)/(3+5)`
`=(12+60)/(8)`
`=(72)/(8)`
`=9`
Thus, the value of x is `9`
2. Find the ratio in which the point `P(11,y)` divides the line segment joining the points `A(15,5)` and `B(9,20)`? Also find the value of y.
Solution:
Method-1 : considering the ratio `m:n`
Let the point `P(11,y)` divides the line segment joining the points `A(15,5)` and `B(9,20)` in the ratio `m:n`
The given points are `A(15,5),B(9,20)`
`:. x_1=15,y_1=5,x_2=9,y_2=20`
Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`
`:.P(11,y)=((m(9)+n(15))/(m+n),(m(20)+n(5))/(m+n))`
`:. (m(9)+n(15))/(m+n)=11`
`:. 9m+15n=11(m+n)`
`:. 9m+15n=11m+11n`
`:. 9m-11m=11n-15n`
`:. -2m=-4n`
`:. 2m=4n`
`:. m/n=(4)/(2)`
`:. m/n=(2)/(1)`
The point divides the line joining `A(15,5)` and `B(9,20)` in the ratio `2:1`
Putting `m=2,n=1`, we get the coordinates of point P
`y=(my_2+ny_1)/(m+n)`
`=(2*20+1*5)/(2+1)`
`=(40+5)/(3)`
`=(45)/(3)`
`=15`
Thus, the value of y is `15`
Method-2 : considering the ratio `k:1`
Let the point `P(11,y)` divides the line segment joining the points `A(15,5)` and `B(9,20)` in the ratio `k:1`
The given points are `A(15,5),B(9,20)`
`:. x_1=15,y_1=5,x_2=9,y_2=20`
Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`
`:.P(11,y)=((k(9)+(15))/(k+1),(k(20)+(5))/(k+1))`
`:. (k(9)+(15))/(k+1)=11`
`:. 9k+15=11(k+1)`
`:. 9k+15=11k+11`
`:. 9k-11k=11-15`
`:. -2k=-4`
`:. 2k=4`
`:. k=(4)/(2)`
`:. k=(2)/(1)`
The point divides the line joining `A(15,5)` and `B(9,20)` in the ratio `2:1`
Putting `k=2`, we get the coordinates of point P
`y=(ky_2+y_1)/(k+1)`
`=(2*20+5)/(2+1)`
`=(40+5)/(3)`
`=(45)/(3)`
`=15`
Thus, the value of y is `15`
3. Find the ratio in which the point `P(-3,y)` divides the line segment joining the points `A(-5,-4)` and `B(-2,3)`? Also find the value of y.
Solution:
Method-1 : considering the ratio `m:n`
Let the point `P(-3,y)` divides the line segment joining the points `A(-5,-4)` and `B(-2,3)` in the ratio `m:n`
The given points are `A(-5,-4),B(-2,3)`
`:. x_1=-5,y_1=-4,x_2=-2,y_2=3`
Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`
`:.P(-3,y)=((m(-2)+n(-5))/(m+n),(m(3)+n(-4))/(m+n))`
`:. (m(-2)+n(-5))/(m+n)=-3`
`:. -2m-5n=-3(m+n)`
`:. -2m-5n=-3m-3n`
`:. -2m+3m=-3n+5n`
`:. m=2n`
`:. m/n=(2)/(1)`
The point divides the line joining `A(-5,-4)` and `B(-2,3)` in the ratio `2:1`
Putting `m=2,n=1`, we get the coordinates of point P
`y=(my_2+ny_1)/(m+n)`
`=(2*3+1*-4)/(2+1)`
`=(6-4)/(3)`
`=(2)/(3)`
Thus, the value of y is `2/3`
Method-2 : considering the ratio `k:1`
Let the point `P(-3,y)` divides the line segment joining the points `A(-5,-4)` and `B(-2,3)` in the ratio `k:1`
The given points are `A(-5,-4),B(-2,3)`
`:. x_1=-5,y_1=-4,x_2=-2,y_2=3`
Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`
`:.P(-3,y)=((k(-2)+(-5))/(k+1),(k(3)+(-4))/(k+1))`
`:. (k(-2)+(-5))/(k+1)=-3`
`:. -2k-5=-3(k+1)`
`:. -2k-5=-3k-3`
`:. -2k+3k=-3+5`
`:. k=2`
`:. k=(2)/(1)`
The point divides the line joining `A(-5,-4)` and `B(-2,3)` in the ratio `2:1`
Putting `k=2`, we get the coordinates of point P
`y=(ky_2+y_1)/(k+1)`
`=(2*3-4)/(2+1)`
`=(6-4)/(3)`
`=(2)/(3)`
Thus, the value of y is `2/3`
4. Find the ratio in which the point `P(-4,y)` divides the line segment joining the points `A(-6,10)` and `B(3,-8)`? Also find the value of y.
Solution:
Method-1 : considering the ratio `m:n`
Let the point `P(-4,y)` divides the line segment joining the points `A(-6,10)` and `B(3,-8)` in the ratio `m:n`
The given points are `A(-6,10),B(3,-8)`
`:. x_1=-6,y_1=10,x_2=3,y_2=-8`
Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`
`:.P(-4,y)=((m(3)+n(-6))/(m+n),(m(-8)+n(10))/(m+n))`
`:. (m(3)+n(-6))/(m+n)=-4`
`:. 3m-6n=-4(m+n)`
`:. 3m-6n=-4m-4n`
`:. 3m+4m=-4n+6n`
`:. 7m=2n`
`:. m/n=(2)/(7)`
The point divides the line joining `A(-6,10)` and `B(3,-8)` in the ratio `2:7`
Putting `m=2,n=7`, we get the coordinates of point P
`y=(my_2+ny_1)/(m+n)`
`=(2*-8+7*10)/(2+7)`
`=(-16+70)/(9)`
`=(54)/(9)`
`=6`
Thus, the value of y is `6`
Method-2 : considering the ratio `k:1`
Let the point `P(-4,y)` divides the line segment joining the points `A(-6,10)` and `B(3,-8)` in the ratio `k:1`
The given points are `A(-6,10),B(3,-8)`
`:. x_1=-6,y_1=10,x_2=3,y_2=-8`
Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`
`:.P(-4,y)=((k(3)+(-6))/(k+1),(k(-8)+(10))/(k+1))`
`:. (k(3)+(-6))/(k+1)=-4`
`:. 3k-6=-4(k+1)`
`:. 3k-6=-4k-4`
`:. 3k+4k=-4+6`
`:. 7k=2`
`:. k=(2)/(7)`
The point divides the line joining `A(-6,10)` and `B(3,-8)` in the ratio `2:7`
Putting `k=2/7`, we get the coordinates of point P
`y=(ky_2+y_1)/(k+1)`
`=(2/7*-8+10)/(2/7+1)`
`=(2*-8+10*7)/(2+7)`
`=(-16+70)/(9)`
`=(54)/(9)`
`=6`
Thus, the value of y is `6`
5. Find the ratio in which the point `P(x,4)` divides the line segment joining the points `A(2,1)` and `B(7,6)`? Also find the value of x.
Solution:
Method-1 : considering the ratio `m:n`
Let the point `P(x,4)` divides the line segment joining the points `A(2,1)` and `B(7,6)` in the ratio `m:n`
The given points are `A(2,1),B(7,6)`
`:. x_1=2,y_1=1,x_2=7,y_2=6`
Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`
`:.P(x,4)=((m(7)+n(2))/(m+n),(m(6)+n(1))/(m+n))`
`:. (m(6)+n(1))/(m+n)=4`
`:. 6m+n=4(m+n)`
`:. 6m+n=4m+4n`
`:. 6m-4m=4n-n`
`:. 2m=3n`
`:. m/n=(3)/(2)`
:. The point divides the line joining `A(2,1)` and `B(7,6)` in the ratio `3:2`
Putting `m=3,n=2`, we get the coordinates of point P
`x=(mx_2+nx_1)/(m+n)`
`=(3*7+2*2)/(3+2)`
`=(21+4)/(5)`
`=(25)/(5)`
`=5`
Thus, the value of x is `5`
Method-2 : considering the ratio `k:1`
Let the point `P(x,4)` divides the line segment joining the points `A(2,1)` and `B(7,6)` in the ratio `k:1`
The given points are `A(2,1),B(7,6)`
`:. x_1=2,y_1=1,x_2=7,y_2=6`
Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`
`:.P(x,4)=((k(7)+(2))/(k+1),(k(6)+(1))/(k+1))`
`:. (k(6)+(1))/(k+1)=4`
`:. 6k+1=4(k+1)`
`:. 6k+1=4k+4`
`:. 6k-4k=4-1`
`:. 2k=3`
`:. k=(3)/(2)`
:. The point divides the line joining `A(2,1)` and `B(7,6)` in the ratio `3:2`
Putting `k=3/2`, we get the coordinates of point P
`x=(kx_2+x_1)/(k+1)`
`=(3/2*7+2)/(3/2+1)`
`=(3*7+2*2)/(3+2)`
`=(21+4)/(5)`
`=(25)/(5)`
`=5`
Thus, the value of x is `5`
6. Find the ratio in which the point `P(x,0)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)`? Also find the value of x.
Solution:
Method-1 : considering the ratio `m:n`
Let the point `P(x,0)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)` in the ratio `m:n`
The given points are `A(2,-4),B(-3,6)`
`:. x_1=2,y_1=-4,x_2=-3,y_2=6`
Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`
`:.P(x,0)=((m(-3)+n(2))/(m+n),(m(6)+n(-4))/(m+n))`
`:. (m(6)+n(-4))/(m+n)=0`
`:. 6m-4n=0`
`:. 6m=4n`
`:. m/n=(4)/(6)`
`:. m/n=(2)/(3)`
:. The point divides the line joining `A(2,-4)` and `B(-3,6)` in the ratio `2:3`
Putting `m=2,n=3`, we get the coordinates of point P
`x=(mx_2+nx_1)/(m+n)`
`=(2*-3+3*2)/(2+3)`
`=(-6+6)/(5)`
`=(0)/(5)`
Thus, the value of x is `0`
Method-2 : considering the ratio `k:1`
Let the point `P(x,0)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)` in the ratio `k:1`
The given points are `A(2,-4),B(-3,6)`
`:. x_1=2,y_1=-4,x_2=-3,y_2=6`
Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`
`:.P(x,0)=((k(-3)+(2))/(k+1),(k(6)+(-4))/(k+1))`
`:. (k(6)+(-4))/(k+1)=0`
`:. 6k-4=0`
`:. 6k=4`
`:. k=(4)/(6)`
`:. k=(2)/(3)`
:. The point divides the line joining `A(2,-4)` and `B(-3,6)` in the ratio `2:3`
Putting `k=2/3`, we get the coordinates of point P
`x=(kx_2+x_1)/(k+1)`
`=(2/3*-3+2)/(2/3+1)`
`=(2*-3+2*3)/(2+3)`
`=(-6+6)/(5)`
`=(0)/(5)`
Thus, the value of x is `0`
7. Find the ratio in which the point `P(0,y)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)`? Also find the value of y.
Solution:
Method-1 : considering the ratio `m:n`
Let the point `P(0,y)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)` in the ratio `m:n`
The given points are `A(2,-4),B(-3,6)`
`:. x_1=2,y_1=-4,x_2=-3,y_2=6`
Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`
`:.P(0,y)=((m(-3)+n(2))/(m+n),(m(6)+n(-4))/(m+n))`
`:. (m(-3)+n(2))/(m+n)=0`
`:. -3m+2n=0`
`:. -3m=-2n`
`:. 3m=2n`
`:. m/n=(2)/(3)`
The point divides the line joining `A(2,-4)` and `B(-3,6)` in the ratio `2:3`
Putting `m=2,n=3`, we get the coordinates of point P
`y=(my_2+ny_1)/(m+n)`
`=(2*6+3*-4)/(2+3)`
`=(12-12)/(5)`
`=(0)/(5)`
Thus, the value of y is `0`
Method-2 : considering the ratio `k:1`
Let the point `P(0,y)` divides the line segment joining the points `A(2,-4)` and `B(-3,6)` in the ratio `k:1`
The given points are `A(2,-4),B(-3,6)`
`:. x_1=2,y_1=-4,x_2=-3,y_2=6`
Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`
`:.P(0,y)=((k(-3)+(2))/(k+1),(k(6)+(-4))/(k+1))`
`:. (k(-3)+(2))/(k+1)=0`
`:. -3k+2=0`
`:. -3k=-2`
`:. 3k=2`
`:. k=(2)/(3)`
The point divides the line joining `A(2,-4)` and `B(-3,6)` in the ratio `2:3`
Putting `k=2/3`, we get the coordinates of point P
`y=(ky_2+y_1)/(k+1)`
`=(2/3*6-4)/(2/3+1)`
`=(2*6-4*3)/(2+3)`
`=(12-12)/(5)`
`=(0)/(5)`
Thus, the value of y is `0`
This material is intended as a summary. Use your textbook for detail explanation.
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