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

1. x
2. length of AP.

Answer 1

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