Question

If a+b+c=11 and ab + bc + ca = 35, find the value of (a-b)²+(b-c)²+(c-a)².

• A. 30
• B. 32
• C. 35
• D. 40
• E. 25

Answer 1

The Correct Option is B

Step 1: Understand the given expression.
We are given the following conditions:
\( a + b + c = 11 \)
\( ab + bc + ca = 35 \)
We need to find the value of the expression: \[ (a - b)^2 + (b - c)^2 + (c - a)^2 \]

Step 2: Expand the given expression.
First, let's expand the terms: \[ (a - b)^2 + (b - c)^2 + (c - a)^2 = a^2 - 2ab + b^2 + b^2 - 2bc + c^2 + c^2 - 2ca + a^2 \] Combine like terms: \[ = 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca \] Factor out the common factor of 2: \[ = 2(a^2 + b^2 + c^2 - ab - bc - ca) \]

Step 3: Use the given conditions.
We are given \( ab + bc + ca = 35 \). Now, we need to find \( a^2 + b^2 + c^2 \). We can use the identity: \[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \] Substitute \( a + b + c = 11 \) and \( ab + bc + ca = 35 \): \[ 11^2 = a^2 + b^2 + c^2 + 2(35) \] \[ 121 = a^2 + b^2 + c^2 + 70 \] \[ a^2 + b^2 + c^2 = 121 - 70 = 51 \]

Step 4: Substitute into the expression.
Now substitute \( a^2 + b^2 + c^2 = 51 \) and \( ab + bc + ca = 35 \) into the expression for \( (a - b)^2 + (b - c)^2 + (c - a)^2 \): \[ = 2(a^2 + b^2 + c^2 - ab - bc - ca) \] \[ = 2(51 - 35) = 2 \times 16 = 32 \]

Step 5: Conclusion.
The value of \( (a - b)^2 + (b - c)^2 + (c - a)^2 \) is 32.

Final Answer:
The correct option is (B): 32.