If a, b, c are distinct +ve real numbers and a^2 + b^2 + c^2 = 1

• Quadratic Equations

If a, b, c are distinct +ve real numbers and a² + b² + c² = 1, then ab + bc + ca is

1. greater than 1
2. equal to 1
3. less than 1
4. any real number

Solution

In such type of problem if sum of the squares of number is known and you need product of numbers taken two at a time or needed range of the product of numbers taken two at a time. Start with square of the sum of the numbers like

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

2(ab + bc + ca) = (a + b + c)² – (a² + b² + c²)

(ab + bc + ca) = [(a + b + c)² – 1]/2

Since a² + b² + c² = 1, all a, b and c are less than 1. So, (ab + bc + ca) must be less than 1.

The correct option is C.