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 1

Let the point P (*x*, 2) divide the line segment joining the points A (12, 5) and B (4, −3) in the ratio *k:*1.

Then, the coordinates of P are `((4k+12)/(k+1),(-3k+5)/(k+1))`

Now, the coordinates of P are (*x*, 2).

`therefore (4k+12)/(k+1)=x and (-3k+5)/(k+1)=2`

`(-3k+5)/(k+1)=2`

`-3k+5=2k+2`

`5k=3`

`k=3/5`

Substituting `k=3/5 " in" (4k+12)/(k+1)=x`

we get

`x=(4xx3/5+12)/(3/5+1)`

`x=(12+60)/(3+5)`

`x=72/8`

x=9

Thus, the value of *x *is 9.

Also, the point P divides the line segment joining the points A(12, 5) and (4, −3) in the ratio 3/5:1 i.e. 3:5.

#### Solution 2

Let k be the ratio in which the point P(x,2) divides the line joining the points

`A(x_1 =12, y_1=5) and B(x_2 = 4, y_2 = -3 ) .` Then

`x= (kxx4+12)/(k+1) and 2 = (kxx (-3)+5) /(k+1)`

Now,

` 2 = (kxx (-3)+5)/(k+1) ⇒ 2k+2 = -3k +5 ⇒ k=3/5`

Hence, the required ratio is3:5 .