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Question
Calculate the ratio in which the line joining A(−4, 2) and B(3, 6) is divided by point P(x, 3). Also, find
- x
- length of AP.
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Solution
Given points are A(–4, 2) and B(3, 6)
Let P(x, 3) divides the line joining A(4, 2) and B(3, 6) in the ratio k : 1.
Thus, we have
`(3k - 4)/(k +1) = x` ...(i)
And `(6k + 2)/(k +1) = 3`
6k + 2 = 3(k + 1)
`=>` 6k + 2 = 3k + 3
`=>` 3k = 1
`=> k= 1/3`
Substituting the value of k in equation (i), we have
i. `(3 xx 1/3 -4)/(1/3 + 1) = x`
`=> (-3)/(4/3) = x`
`=> (-9)/4 = x`
Therefore, coordinates of P are `(-9/4 , 3)`
ii. Now let us find the distance AP
`AP = sqrt((-9/4 +4)^2 + (3 - 2)^2)`
= `sqrt((7/4)^2 + (1)^2)`
= `sqrt(49/16 + 1)`
= `sqrt((49 + 16)/16)`
= `sqrt(65/16)`
`=> AP = sqrt(65)/4` units
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