Question

If area of triangle is 35 sq. units with vertices \( (2,-6), (5,4) \) and \( (k,4) \), then \( k \) is _____

• A. \( 12 \)
• B. \( -12, -2 \)
• C. \( -2 \)
• D. \( 12, -2 \)

Hint:

Always use determinant formula for area of triangle in coordinate geometry.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ can be found using the determinant formula: $\text{Area} = \pm \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$. Since area is given, we must consider both positive and negative values for the determinant.
Step 2: Formula Application:
$35 = \frac{1}{2} |2(4 - 4) + 5(4 - (-6)) + k(-6 - 4)|$ $70 = |2(0) + 5(10) + k(-10)|$ $70 = |50 - 10k|$
Step 3: Explanation:
Case 1: $50 - 10k = 70 \implies -10k = 20 \implies k = -2$ Case 2: $50 - 10k = -70 \implies -10k = -120 \implies k = 12$
Step 4: Final Answer:
The values of $k$ are $12, -2$.