Question

Find the ratio in which the sphere x² + y² + z²= 504 divides the line joining the point (12, -4, 8) and (27, -9, 18).

Answer 1

Given: A(12, -4, 8) and B(27, -9, 18) 

To find: the ratio in which the line segment AB is divided by the sphere x² + y² + z² = 504 

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,

Let C(x, y, z) be any point on given plane and C divides AB in ratio k: 1

Therefore, m = k and n = 1 

A(12, -4, 8) and B(27, -9, 18) 

Coordinates of C using section formula:

⇒ k² = \(\frac{20}{45}\)

⇒ k² = \(\frac{4}{9}\)

⇒  k = \(\frac{2}{3}\) 

Hence, the sphere divides AB in ratio 2 : 3