# A point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10). Find the coordinates.

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A point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10). Find the coordinates.

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Given: A point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10)

To find: the coordinates of C

Formula used:

Section Formula:

A line AB is divided by C in m:n where A(x, y, z) and B(a, b, c).

The coordinates of C is given by,

$\Big(\frac{nx+ma}{m+n},\frac{ny+mb}{m+n},\frac{nz+mc}{m+n}\Big)$

Let Point C(x, y, 8), and C divides AB in ratio k: 1

Therefore, m = k and n = 1

A (2, -3, 4) and B (8, 0, 10)

⇒ (x, y, 8) = $\Big(\frac{k(8)+1(2)}{k+1},\frac{k(0)+1(-3)}{k+1},\frac{k(10)+1(4)}{k+1}\Big)$

⇒ (x, y, 8) = $\Big(\frac{8k+2}{k+1},\frac{-3}{k+1}, \frac{10k+4}{k+1}\Big)$

On comparing:

$\frac{10k+4}{k+1}$ = 8

⇒ 10k + 4 = 8(k + 1)

⇒ 10k + 4 = 8k + 8

⇒ 10k – 8k = 8 – 4

⇒ 2k = 4

⇒ k = $\frac{4}{2}$

⇒ k = 2

Here C divides AB in ratio 2:1

⇒ y = -1

Hence, Coordinates of C are (6, -1, 8)

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