Show that the points A \left(\right. 3 , 2 , - 4 \left.\right) , B \left(\right. 5 , 4 , - 6 \left.\right) and C \left(\right. 9 , 8 , - 10 \left.\right) are collinear, find the ratio in which B divides \overset{\overline}{A C}.

Engineering

Mathematics

Line of Greatest Slope

Question

Show that the points

A (3, 2, - 4), B (5, 4, - 6)

and

C (9, 8, - 10)

are collinear, find the ratio in which

B

divides

\overline{A C}

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Given that

Point A(3,2,-4), B(5,4,-6) & C(9,8,-10) are collinear

B must divide line segment AC in some ratio externally and internally

We know that

Co-ordinate of point

A (x, y, z)

that divides line segment joining

(x_{1}, y_{1}, z_{1})

and

(x_{2}, y_{2}, z_{2})

in ratio

m : n

(x, y, z) = (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})

Let point

B (5, 4, - 6)

divide line segment

A (3, 2, - 4)

and

C (9, 8, - 10)

in the ratio

k : 1

B (5, 4, - 6) = (\frac{9 k + 3}{k + 1}, \frac{8 k + 2}{k + 1}, \frac{- 10 k - 4}{k + 1})

Comparing x-coordinate of B

5 = \frac{9 k + 3}{k + 1}

5 k + 5 = 9 k + 3

9 k - 5 k = 5 - 3

4 k = 2

k = \frac{1}{2}

So,

k : 1 = 1 : 2

Thus, point B divides AC in the ratio 1:2

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