1. Find the ratio in which the point `P(3/4,5/12)` divides the line segment joining the points `A(1/2,3/2)` and `B(2,-5)`

Solution:
Method-1 : considering the ratio `m:n`

Suppose `P(3/4,5/12)` divides the line joining `A(1/2,3/2)` and `B(2,-5)` in the ratio `m:n`

Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`

`:.P(3/4,5/12)=((m(2)+n(1/2))/(m+n),(m(-5)+n(3/2))/(m+n))`

`:.(m(2)+n(1/2))/(m+n)=3/4` and `(m(-5)+n(3/2))/(m+n)=5/12`

Now, solving first
`:. (m(2)+n(1/2))/(m+n)=3/4`

`:. 4(2m+(n)/(2))=3(m+n)`

`:. 8m+2n=3m+3n`

`:. 8m-3m=3n-2n`

`:. 5m=n`

`:. m/n=(1)/(5)`

`:.` The point `P(3/4,5/12)` divides the line joining `A(1/2,3/2)` and `B(2,-5)` in the ratio `1:5`

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Method-2 : considering the ratio `k:1`

Suppose `P(3/4,5/12)` divides the line joining `A(1/2,3/2)` and `B(2,-5)` in the ratio `k:1`

Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`

`:.P(3/4,5/12)=((k(2)+(1/2))/(k+1),(k(-5)+(3/2))/(k+1))`

`:.(k(2)+(1/2))/(k+1)=3/4` and `(k(-5)+(3/2))/(k+1)=5/12`

Now, solving first
`:. (k(2)+(1/2))/(k+1)=3/4`

`:. 4(2k+1/2)=3(k+1)`

`:. 8k+2=3k+3`

`:. 8k-3k=3-2`

`:. 5k=1`

`:. k=(1)/(5)`

`:.` The point `P(3/4,5/12)` divides the line joining `A(1/2,3/2)` and `B(2,-5)` in the ratio `1:5`

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2. Find the ratio in which the point `P(-1,6)` divides the line segment joining the points `A(3,10)` and `B(6,-8)`

Solution:
Method-1 : considering the ratio `m:n`

Suppose `P(-1,6)` divides the line joining `A(3,10)` and `B(6,-8)` in the ratio `m:n`

Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`

`:.P(-1,6)=((m(6)+n(3))/(m+n),(m(-8)+n(10))/(m+n))`

`:.(m(6)+n(3))/(m+n)=-1` and `(m(-8)+n(10))/(m+n)=6`

Now, solving first
`:. (m(6)+n(3))/(m+n)=-1`

`:. 6m+3n=-1(m+n)`

`:. 6m+3n=-m-n`

`:. 6m+m=-n-3n`

`:. 7m=-4n`

`:. m/n=(-4)/(7)`

As the ratio is negative, the point `P(-1,6)` divides the line joining `A(3,10)` and `B(6,-8)` externally in the ratio `4:7`

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Method-2 : considering the ratio `k:1`

Suppose `P(-1,6)` divides the line joining `A(3,10)` and `B(6,-8)` in the ratio `k:1`

Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`

`:.P(-1,6)=((k(6)+(3))/(k+1),(k(-8)+(10))/(k+1))`

`:.(k(6)+(3))/(k+1)=-1` and `(k(-8)+(10))/(k+1)=6`

Now, solving first
`:. (k(6)+(3))/(k+1)=-1`

`:. 6k+3=-1(k+1)`

`:. 6k+3=-k-1`

`:. 6k+k=-1-3`

`:. 7k=-4`

`:. k=(-4)/(7)`

As the ratio is negative, the point `P(-1,6)` divides the line joining `A(3,10)` and `B(6,-8)` externally in the ratio `4:7`

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3. Find the ratio in which the point `P(-2,3)` divides the line segment joining the points `A(-3,5)` and `B(4,-9)`

Solution:
Method-1 : considering the ratio `m:n`

Suppose `P(-2,3)` divides the line joining `A(-3,5)` and `B(4,-9)` in the ratio `m:n`

Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`

`:.P(-2,3)=((m(4)+n(-3))/(m+n),(m(-9)+n(5))/(m+n))`

`:.(m(4)+n(-3))/(m+n)=-2` and `(m(-9)+n(5))/(m+n)=3`

Now, solving first
`:. (m(4)+n(-3))/(m+n)=-2`

`:. 4m-3n=-2(m+n)`

`:. 4m-3n=-2m-2n`

`:. 4m+2m=-2n+3n`

`:. 6m=n`

`:. m/n=(1)/(6)`

`:.` The point `P(-2,3)` divides the line joining `A(-3,5)` and `B(4,-9)` in the ratio `1:6`

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Method-2 : considering the ratio `k:1`

Suppose `P(-2,3)` divides the line joining `A(-3,5)` and `B(4,-9)` in the ratio `k:1`

Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`

`:.P(-2,3)=((k(4)+(-3))/(k+1),(k(-9)+(5))/(k+1))`

`:.(k(4)+(-3))/(k+1)=-2` and `(k(-9)+(5))/(k+1)=3`

Now, solving first
`:. (k(4)+(-3))/(k+1)=-2`

`:. 4k-3=-2(k+1)`

`:. 4k-3=-2k-2`

`:. 4k+2k=-2+3`

`:. 6k=1`

`:. k=(1)/(6)`

`:.` The point `P(-2,3)` divides the line joining `A(-3,5)` and `B(4,-9)` in the ratio `1:6`

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4. Find the ratio in which the point `P(3,10)` divides the line segment joining the points `A(5,12)` and `B(2,9)`

Solution:
Method-1 : considering the ratio `m:n`

Suppose `P(3,10)` divides the line joining `A(5,12)` and `B(2,9)` in the ratio `m:n`

Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`

`:.P(3,10)=((m(2)+n(5))/(m+n),(m(9)+n(12))/(m+n))`

`:.(m(2)+n(5))/(m+n)=3` and `(m(9)+n(12))/(m+n)=10`

Now, solving first
`:. (m(2)+n(5))/(m+n)=3`

`:. 2m+5n=3(m+n)`

`:. 2m+5n=3m+3n`

`:. 2m-3m=3n-5n`

`:. -m=-2n`

`:. m=2n`

`:. m/n=(2)/(1)`

`:.` The point `P(3,10)` divides the line joining `A(5,12)` and `B(2,9)` in the ratio `2:1`

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Method-2 : considering the ratio `k:1`

Suppose `P(3,10)` divides the line joining `A(5,12)` and `B(2,9)` in the ratio `k:1`

Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`

`:.P(3,10)=((k(2)+(5))/(k+1),(k(9)+(12))/(k+1))`

`:.(k(2)+(5))/(k+1)=3` and `(k(9)+(12))/(k+1)=10`

Now, solving first
`:. (k(2)+(5))/(k+1)=3`

`:. 2k+5=3(k+1)`

`:. 2k+5=3k+3`

`:. 2k-3k=3-5`

`:. -k=-2`

`:. k=2`

`:. k=(2)/(1)`

`:.` The point `P(3,10)` divides the line joining `A(5,12)` and `B(2,9)` in the ratio `2:1`

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5. Find the ratio in which the point `P(6,17)` divides the line segment joining the points `A(1,-3)` and `B(3,5)`

Solution:
Method-1 : considering the ratio `m:n`

Suppose `P(6,17)` divides the line joining `A(1,-3)` and `B(3,5)` in the ratio `m:n`

Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`

`:.P(6,17)=((m(3)+n(1))/(m+n),(m(5)+n(-3))/(m+n))`

`:.(m(3)+n(1))/(m+n)=6` and `(m(5)+n(-3))/(m+n)=17`

Now, solving first
`:. (m(3)+n(1))/(m+n)=6`

`:. 3m+n=6(m+n)`

`:. 3m+n=6m+6n`

`:. 3m-6m=6n-n`

`:. -3m=5n`

`:. 3m=-5n`

`:. m/n=(-5)/(3)`

As the ratio is negative, the point `P(6,17)` divides the line joining `A(1,-3)` and `B(3,5)` externally in the ratio `5:3`

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Method-2 : considering the ratio `k:1`

Suppose `P(6,17)` divides the line joining `A(1,-3)` and `B(3,5)` in the ratio `k:1`

Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`

`:.P(6,17)=((k(3)+(1))/(k+1),(k(5)+(-3))/(k+1))`

`:.(k(3)+(1))/(k+1)=6` and `(k(5)+(-3))/(k+1)=17`

Now, solving first
`:. (k(3)+(1))/(k+1)=6`

`:. 3k+1=6(k+1)`

`:. 3k+1=6k+6`

`:. 3k-6k=6-1`

`:. -3k=5`

`:. 3k=-5`

`:. k=(-5)/(3)`

As the ratio is negative, the point `P(6,17)` divides the line joining `A(1,-3)` and `B(3,5)` externally in the ratio `5:3`

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6. Find the ratio in which the point `P(12,23)` divides the line segment joining the points `A(2,8)` and `B(6,14)`

Solution:
Method-1 : considering the ratio `m:n`

Suppose `P(12,23)` divides the line joining `A(2,8)` and `B(6,14)` in the ratio `m:n`

Using section formula
`P(x,y)=((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n))`

`:.P(12,23)=((m(6)+n(2))/(m+n),(m(14)+n(8))/(m+n))`

`:.(m(6)+n(2))/(m+n)=12` and `(m(14)+n(8))/(m+n)=23`

Now, solving first
`:. (m(6)+n(2))/(m+n)=12`

`:. 6m+2n=12(m+n)`

`:. 6m+2n=12m+12n`

`:. 6m-12m=12n-2n`

`:. -6m=10n`

`:. 6m=-10n`

`:. m/n=(-10)/(6)`

`:. m/n=(-5)/(3)`

As the ratio is negative, the point `P(12,23)` divides the line joining `A(2,8)` and `B(6,14)` externally in the ratio `5:3`

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Method-2 : considering the ratio `k:1`

Suppose `P(12,23)` divides the line joining `A(2,8)` and `B(6,14)` in the ratio `k:1`

Using section formula
`P(x,y)=((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1))`

`:.P(12,23)=((k(6)+(2))/(k+1),(k(14)+(8))/(k+1))`

`:.(k(6)+(2))/(k+1)=12` and `(k(14)+(8))/(k+1)=23`

Now, solving first
`:. (k(6)+(2))/(k+1)=12`

`:. 6k+2=12(k+1)`

`:. 6k+2=12k+12`

`:. 6k-12k=12-2`

`:. -6k=10`

`:. 6k=-10`

`:. k=(-10)/(6)`

`:. k=(-5)/(3)`

As the ratio is negative, the point `P(12,23)` divides the line joining `A(2,8)` and `B(6,14)` externally in the ratio `5:3`