Question

If the area of the triangle formed by the lines \( y = x + c \) and \( 2x^2 + 5xy + 3y^2 = 0 \) is \( \frac{1}{20} \) sq. units, then \( c = \)

• A. \( \pm 1 \)
• B. \( \pm \sqrt{2} \)
• C. \( \pm 3 \)
• D. \( \pm \sqrt{3} \)

Hint:

When solving for constants in geometric problems, always check if the area or distance condition can be used to form a solvable equation.

Answer 1

The Correct Option is A

We are given the equation of two lines: \( y = x + c \) and \( 2x^2 + 5xy + 3y^2 = 0 \). The area of the triangle formed by these lines is given as \( \frac{1}{20} \) square units.
Step 1: First, express the area of a triangle formed by two lines using the formula for the area of the triangle formed by two intersecting lines. The formula for the area is \( \frac{1}{2} \left| \text{determinant of the coefficients of the lines} \right| \).
Step 2: Substitute the given lines into the formula and calculate the determinant. Use the given area condition to solve for \( c \).
After simplifying, we find that \( c = \pm 1 \).
Thus, the correct answer is \( \pm 1 \).