Question

Given that \( \cot \left( \frac{A+B}{2} \right) \cdot \tan \left( \frac{A-B}{2} \right) = \), and the equation \( \frac{x}{2} + \frac{y}{3} + \frac{2}{6} - 1 = 0 \), find the area of \( \Delta ABC = 2 \).

• A. \( 2 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( 5 \)

Hint:

When given multiple types of equations (trigonometric and coordinate), try simplifying each to extract relevant information. In this case, the area was directly provided.

Answer 1

The Correct Option is A

We are given a geometric problem that involves both trigonometric functions and coordinate geometry.
Step 1: Analyze the given equations We have the equation: \[ \frac{x}{2} + \frac{y}{3} + \frac{2}{6} - 1 = 0 \] Simplifying the equation: \[ \frac{x}{2} + \frac{y}{3} + \frac{1}{3} = 1 \] Multiplying through by 6 to clear the denominators: \[ 3x + 2y + 2 = 6 \] Simplifying further: \[ 3x + 2y = 4 \] This gives a linear equation involving \( x \) and \( y \), likely corresponding to a line equation in the coordinate plane.
Step 2: Relating the trigonometric functions to the area We are also given the trigonometric equation: \[ \cot \left( \frac{A+B}{2} \right) \cdot \tan \left( \frac{A-B}{2} \right) \] This equation could relate to the angles of the triangle. Typically, this type of relation can help find the area of the triangle through trigonometric identities. However, we are already told that the area of \( \Delta ABC \) is \( 2 \), and thus we do not need to solve for it explicitly.
Step 3: Conclusion Thus, the area of the triangle \( \Delta ABC \) is \( \boxed{2} \).