Question

Given points A (1, 2/3), B (x, -4/5), and C (-1/2, 4) determine the value of x such that all three points are collinear

Answer 1

Explanation:

Let B divides AB in the ratio K:1

`x=(nx1+mx2)/(m+n) \\y=(ny1+my2)/(m+n) \\(-4)/(5) =(1*(2)/(3)+k*4)/(k+1) \\-4k-4=(10)/(3) +20k\\-12 k-12=10+60k\\72k=-22\\36k=-11\\k=-(11)/(36) \\`

so B divides AB in the ratio 11:-36

`x=(-36*1+11 *(-1)/(2) )/(11-36) \\x=(83)/(50)`

Answer 2

`\large \boxed{1.66}`

Explanation:

1. Calculate the equation of the straight line joining A and C.

The equation for a straight line is

y = mx + b

where m is the slope of the line and b is the y-intercept.

The line passes through the points (-½, 4) and (1, ⅔)

(a) Calculate the slope of the line

`\begin{array}{rcl}m & = & (y_(2) - y_(1))/(x_(2) - x_(1))\\\\ & = & ((2)/(3) - 4)/(1 - (-(1)/(2)))\\\\& = & (-(10)/(3))/((3)/(2))\\\\& = & (-10)/(3)*{(2)/(3)}\\\\& = & (-20)/(9)\\\\\end{array}`

(b) Find the y-intercept

Insert the coordinates of one of the points into the equation

`\begin{array}{rcl}y & = & mx + b\\4 & = & (-20)/(9)\left(-(1)/(2)\right) + b \\\\4 & = & (10)/(9) + b\\\\b & = & (36)/(9) - (10)/(9)\\\\b & = & (26)/(9)\\\\\end{array}`

(c) Write the equation for the line

`y = -(20)/(9)x + (26)/(9)`

2. Calculate the value of x when y = -⅘

`\begin{array}{rcl}y & = & -(20)/(9)x + (26)/(9)\\\\-(4)/(5) & = & -(20)/(9)x+ (26)/(9)\\\\36 & = & 100x -130\\100x & = & 166\\x & = & 1.66\\\end{array}\\\text{The value of x is $\large \boxed{\mathbf{1.66}}$}`

The graph below shows your three collinear points.