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Question
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 .
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