If a² + b² + c² + 189 = 6 (a + 4b + 2c), then {ab + bc + ca} is equal to:

Correct Answer is : 3143{14}314

Correct Answer is : 3143{14}314

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RRB ALP CBT I 17 Feb 2026 Shift 2 Paper

Question : 2 of 75

Marks: +1, -0

a^2 + b^2 + c^2 + 189 = 6 (a + 4b + 2c)

\sqrt{ab + bc + ca}

is equal to:

Solution:

Concept:

For an equation of the form

a^{2} + b^{2} + c^{2} + k = 2(ma + nb + pc)

, completing the square gives

a = m

b = n

c = p

Explanation:

Given:

a^{2} + b^{2} + c^{2} + 189 = 6(a + 4b + 2c)

Expand the right-hand side:

6(a + 4b + 2c) = 6a + 24b + 12c

Rearrange:

a^{2} - 6a + b^{2} - 24b + c^{2} - 12c + 189 = 0

Complete the squares:

(a^{2} - 2 \times 3a + 3^{2}) + (b^{2} - 2 \times 12b + 12^{2}) + (c^{2} - 2 \times 6c + 6^{2}) + 189 - 9 - 144 - 36 = 0

This simplifies to:

(a - 3)^{2} + (b - 12)^{2} + (c - 6)^{2} = 0

Since the sum of squares is zero, each square must be zero:

a = 3

b = 12

c = 6

Now compute:

\sqrt{ab + bc + ca} = \sqrt{3 \times 12 + 12 \times 6 + 6 \times 3}

So,

\sqrt{36 + 72 + 18} = \sqrt{126} = \sqrt{9 \times 14} = 3\sqrt{14}

Answer:

\sqrt{ab + bc + ca} = 3\sqrt{14}

, which is option D.

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