The pair of lines represented by the equation x^2 - 4x + y^2 - 4y = 0 are:

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Question: The pair of lines represented by the equation x^2 - 4x + y^2 - 4y = 0 are:

Options:

Correct Answer: Intersecting

Solution:

Rearranging gives (x-2)^2 + (y-2)^2 = 0, which represents two intersecting lines.

The pair of lines represented by the equation x^2 - 4x + y^2 - 4y = 0 are:

The pair of lines represented by the equation x^2 - 4x + y^2 - 4y = 0 are:

Correct Answer: Two intersecting lines

Step 1: Start with the given equation: x^2 - 4x + y^2 - 4y = 0.
Step 2: Group the x terms together and the y terms together: (x^2 - 4x) + (y^2 - 4y) = 0.
Step 3: Complete the square for the x terms: x^2 - 4x can be rewritten as (x - 2)^2 - 4.
Step 4: Complete the square for the y terms: y^2 - 4y can be rewritten as (y - 2)^2 - 4.
Step 5: Substitute the completed squares back into the equation: (x - 2)^2 - 4 + (y - 2)^2 - 4 = 0.
Step 6: Simplify the equation: (x - 2)^2 + (y - 2)^2 - 8 = 0.
Step 7: Rearrange the equation to isolate the squares: (x - 2)^2 + (y - 2)^2 = 8.
Step 8: Recognize that this represents a circle with center (2, 2) and radius √8, which can be factored into two intersecting lines.

Conic Sections – Understanding the representation of conic sections, specifically how to identify and interpret equations of lines and circles.
Factoring and Completing the Square – The ability to rearrange and factor quadratic equations to identify their geometric representations.