Question

The length of x-intercept made by the pair of lines \( 2x^2 + xy - 6y^2 - 2x + 17y - 12 = 0 \) is:

• A. \( 2 \)
• B. \( 10 \)
• C. \( 5 \)
• D. \( 20 \)

Hint:

To find the x-intercept of a pair of lines given by \( Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 \), set \( y = 0 \) and solve for \( x \).

Answer 1

The Correct Option is C

Step 1: Find the points where the pair of lines intersect the x-axis To determine the x-intercepts, we set \( y = 0 \) in the given equation: \[ 2x^2 + xy - 6y^2 - 2x + 17y - 12 = 0 \] \[ \Rightarrow 2x^2 - 2x - 12 = 0 \]

Step 2: Solve the quadratic equation The quadratic equation simplifies to: \[ 2x^2 - 2x - 12 = 0 \] Dividing throughout by 2: \[ x^2 - x - 6 = 0 \] Factoring: \[ (x - 3)(x + 2) = 0 \] \[ \Rightarrow x = 3 \quad \text{or} \quad x = -2 \]
Step 3: Find the x-intercept length The length of the x-intercept is: \[ |3 - (-2)| = |3 + 2| = 5 \] Thus, the required length is: \[ \boxed{5} \]