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Question
Find the ratio in which the point P(m, 6) divides the join of A(-4, 3) and B(2, 8) Also, find the value of m.
Solution
Let the point P(m,6) divide the line AB in the ratio k :.1
Then, by the section formula:
`x = (mx_2+nx_1)/(m+n) , y =(my_2+ny_1)/(m+n)`
The coordinates of P are (m,6).
`m = (2k-4)/(k+1) , 6 = (8k+3)/(k+1)`
⇒ m (k+1)= 2k-4,6k+6=8k+3
⇒m (k+1) = 2k -4 , 6-3= 8k-6k
⇒m(k+1) = 2k-4, 2k = 3
`⇒m(k+1) = 2k-4,k=3/2`
Therefore, the point P divides the line AB in the ratio 3:2
Now, putting the value of k in the equation m(k+1) = 2k-4 , we get:
`m(3/2+1) = 2(3/2)-4`
`⇒ m((3+2)/2) = 3-4`
` ⇒ (5m)/2 = -1 ⇒ 5m = -2 ⇒m=-2/5`
Therefore, the value of `m = -2/5`
So, the coordinates of P are `(-2/5,6).`
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