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Determine the nature of the lines represented by the equation 7x^2 + 2xy + 3y^2
Determine the nature of the lines represented by the equation 7x^2 + 2xy + 3y^2 = 0.
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Practice Questions
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Q1
Determine the nature of the lines represented by the equation 7x^2 + 2xy + 3y^2 = 0.
Parallel
Intersecting
Coincident
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The discriminant indicates that the lines intersect at two distinct points.
Questions & Step-by-step Solutions
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Q: Determine the nature of the lines represented by the equation 7x^2 + 2xy + 3y^2 = 0.
Solution:
The discriminant indicates that the lines intersect at two distinct points.
Steps: 9
Show Steps
Step 1: Identify the equation given, which is 7x^2 + 2xy + 3y^2 = 0.
Step 2: Recognize that this is a quadratic equation in two variables (x and y).
Step 3: To determine the nature of the lines, we need to calculate the discriminant (D) of the equation.
Step 4: The discriminant for a general conic section Ax^2 + Bxy + Cy^2 = 0 is given by D = B^2 - 4AC.
Step 5: In our equation, A = 7, B = 2, and C = 3.
Step 6: Substitute these values into the discriminant formula: D = (2)^2 - 4(7)(3).
Step 7: Calculate D: D = 4 - 84 = -80.
Step 8: Analyze the value of D: Since D < 0, this indicates that the conic does not represent real lines.
Step 9: Conclude that the equation represents two complex lines that do not intersect in the real plane.
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