In a `Delta ABC, angle BCA = 60^@ and AB^2 = BC^2 + CA^2+ X`. What is the value of X?

Options

A. {BC)[CA)
B. `-`{BC){CA)
C. {AB)(BC)
D. Zero

Answer

To solve the problem, we will use the cosine rule in triangle ABC. The cosine rule states that for any triangle with sides a, b, and c opposite to angles A, B, and C respectively, the following relationship holds:

\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

In our case, we have:

- \( A = BCA = 60^\circ \)
- \( a = BC \)
- \( b = CA \)
- \( c = AB \)

According to the problem, we have:

\[ AB^2 = BC^2 + CA^2 + X \]

From the cosine rule, we can express \( AB^2 \) as:

\[ AB^2 = BC^2 + CA^2 - 2 \cdot BC \cdot CA \cdot \cos(60^\circ) \]

Since \( \cos(60^\circ) = \frac{1}{2} \), we can substitute this into the equation:

\[ AB^2 = BC^2 + CA^2 - 2 \cdot BC \cdot CA \cdot \frac{1}{2} \]

This simplifies to:

\[ AB^2 = BC^2 + CA^2 - BC \cdot CA \]

Now, we can equate this with the expression given in the problem:

\[ BC^2 + CA^2 + X = BC^2 + CA^2 - BC \cdot CA \]

To find \( X \), we can rearrange the equation:

\[ X = - BC \cdot CA \]

Thus, the value of \( X \) is:

\[ X = - BC \cdot CA \]

### Summary of Steps:
1. Identify the triangle and the given angle.
2. Apply the cosine rule to express \( AB^2 \).
3. Substitute \( \cos(60^\circ) \) into the cosine rule equation.
4. Rearrange the equation to solve for \( X \).