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)
