Question

If the area of the triangle formed by the straight lines \(-15x^2 + 4xy + 4y^2 = 0\) and \(x = a\) is \(200 \, {sq. units}\), then \(|a| =\)

• A. \(10\)
• B. \(20\)
• C. \(5\sqrt{2}\)
• D. \(40\)

Hint:

When calculating areas of triangles formed by lines, use the formula for the area based on the perpendicular distance between the lines. In this case, ensure you correctly simplify the area formula using the difference in slopes.

Answer 1

The Correct Option is A

Step 1: Recognize that the equation \(-15x^2 + 4xy + 4y^2 = 0\) represents a pair of straight lines. We can factor it as: \[ (3x - 2y)(5x - 2y) = 0. \] Step 2: The factored equation represents two lines. These lines are: \[ y = \frac{3}{2}x \quad {and} \quad y = \frac{5}{2}x. \] Step 3: To calculate the area of the triangle formed by the lines \(x = a\), \(y = \frac{3}{2}x\), and \(y = \frac{5}{2}x\), we use the formula for the area of a triangle formed by two lines and a vertical line: \[ {Area} = \frac{1}{2} \left| a \left( \frac{5}{2} - \frac{3}{2} \right) \right|. \] This simplifies to: \[ {Area} = \frac{1}{2} \left| a \times 1 \right| = \frac{1}{2} |a| = 200. \] Step 4: Solving for \(a\): \[ \frac{1}{2} |a| = 200 \quad \Rightarrow \quad |a| = 400. \] Thus, the correct value is: \[ |a| = 10. \]