If the lines represented by the equation 3x^2 + 4xy + 2y^2 = 0 intersect at the

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Question: If the lines represented by the equation 3x^2 + 4xy + 2y^2 = 0 intersect at the origin, what is the product of their slopes?

Options:

Correct Answer: -2/3

Solution:

The product of the slopes of the lines represented by ax^2 + bxy + cy^2 = 0 is given by c/a. Here, c = 2 and a = 3, so the product is 2/3.

If the lines represented by the equation 3x^2 + 4xy + 2y^2 = 0 intersect at the origin, what is the product of their slopes?

If the lines represented by the equation 3x^2 + 4xy + 2y^2 = 0 intersect at the origin, what is the product of their slopes?

Step 1: Identify the equation given, which is 3x^2 + 4xy + 2y^2 = 0.
Step 2: Recognize that this equation represents two lines that intersect at the origin.
Step 3: Understand that the general form of a conic section can be expressed as ax^2 + bxy + cy^2 = 0.
Step 4: Identify the coefficients from the equation: a = 3, b = 4, and c = 2.
Step 5: Use the formula for the product of the slopes of the lines, which is c/a.
Step 6: Substitute the values of c and a into the formula: c = 2 and a = 3.
Step 7: Calculate the product of the slopes: 2/3.

Quadratic Equations in Two Variables – Understanding how to interpret and manipulate equations of the form ax^2 + bxy + cy^2 = 0, which represent conic sections, specifically lines in this case.
Slope of Lines – Knowledge of how to find the slopes of lines represented by a quadratic equation and the relationship between the coefficients and the slopes.
Product of Slopes – The concept that the product of the slopes of the lines can be derived from the coefficients of the quadratic equation.