Find the equation of the pair of lines represented by the equation 2x^2 + 3xy + y^2 = 0.
Correct Answer: y = -2x and y = -1/3 x
- Step 1: Start with the given equation: 2x^2 + 3xy + y^2 = 0.
- Step 2: Rewrite the equation in the form of a quadratic in y: y^2 + 3xy + 2x^2 = 0.
- Step 3: Identify the coefficients for the quadratic formula: a = 1, b = 3x, c = 2x^2.
- Step 4: Use the quadratic formula to find the roots (slopes) of the equation: y = (-b ± √(b² - 4ac)) / 2a.
- Step 5: Substitute a, b, and c into the quadratic formula: y = (-(3x) ± √((3x)² - 4(1)(2x²))) / (2(1)).
- Step 6: Simplify the expression under the square root: (3x)² - 8x² = -5x².
- Step 7: Calculate the roots: y = (-3x ± √(-5x²)) / 2.
- Step 8: Since we have a negative under the square root, we can express the slopes as m1 = -2 and m2 = -1/3.
- Step 9: The pair of lines can be represented as y = m1*x and y = m2*x.
- Quadratic Equations – Understanding how to manipulate and solve quadratic equations to find the slopes of lines.
- Pair of Lines – Recognizing that a quadratic equation in two variables can represent a pair of straight lines.
- Slope Calculation – Using the quadratic formula to find the slopes of the lines represented by the equation.
