Question: The condition for the lines represented by the equation x^2 + y^2 + 2xy = 0 to be coincident is:
Options:
Discriminant = 0
Discriminant > 0
Discriminant < 0
None of the above
Correct Answer: Discriminant = 0
Solution:
For the lines to be coincident, the discriminant of the quadratic must be zero.
The condition for the lines represented by the equation x^2 + y^2 + 2xy = 0 to b
Practice Questions
Q1
The condition for the lines represented by the equation x^2 + y^2 + 2xy = 0 to be coincident is:
Discriminant = 0
Discriminant > 0
Discriminant < 0
None of the above
Questions & Step-by-Step Solutions
The condition for the lines represented by the equation x^2 + y^2 + 2xy = 0 to be coincident is:
Step 1: Start with the given equation: x^2 + y^2 + 2xy = 0.
Step 2: Recognize that this equation can be rearranged into a standard quadratic form in terms of y: y^2 + 2xy + x^2 = 0.
Step 3: Identify the coefficients of the quadratic equation: a = 1 (coefficient of y^2), b = 2x (coefficient of y), c = x^2 (constant term).
Step 4: Recall the formula for the discriminant of a quadratic equation, which is D = b^2 - 4ac.
Step 5: Substitute the coefficients into the discriminant formula: D = (2x)^2 - 4(1)(x^2).
Step 6: Simplify the discriminant: D = 4x^2 - 4x^2 = 0.
Step 7: Conclude that for the lines to be coincident, the discriminant must equal zero, which we have shown is true.
Quadratic Equations – Understanding the properties of quadratic equations, particularly the condition for coincident lines, which involves the discriminant.
Discriminant – The discriminant of a quadratic equation determines the nature of the roots; a discriminant of zero indicates that the roots are real and coincident.
Lines in Geometry – The concept of coincident lines in geometry, where two lines occupy the same space and have infinite points in common.
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