Question

If b is the mean proportional of a and c then the mean proportional of a² + b² and b² + c² is ______.

Options

• a + b + c

• ab + cа

• ca + cb

• ab + bc

Answer 1

If b is the mean proportional of a and c then the mean proportional of a² + b² and b² + c² is ab + bc.

Explanation:

Let x be the mean proportional between a² + b² and b² + c²

Then,

a² + b² : x : : x : b² + c²

`(a^2 + b^2)/x = x/(b^2 + c^2)`

x² = (a² + b²) (b² + c²)

x² = a²b² + a²c² + b⁴ + b²c²

Given b is the mean proportional of a and c

b² = ac

Substitute:

x² = a²(ac) + a²c² + (ac)² + (ac)c²

x² = a³c + 2a²c² + ac³

x² = ac(a² + 2ac + c²)

x² = ac(a + c)²

x = `sqrt(ac) (a + c)`

x = b(a + c)

x = ab + bc