The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 9 = 0 are:

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Question: The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 9 = 0 are:

Options:

Correct Answer: Intersecting

Solution:

Rearranging gives (x-2)^2 + (y-3)^2 = 0, which represents a single point, hence the lines are coincident.

The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 9 = 0 are:

The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 9 = 0 are:

Step 1: Start with the given equation: x^2 - 4x + y^2 - 6y + 9 = 0.
Step 2: Rearrange the equation to group the x and y terms: (x^2 - 4x) + (y^2 - 6y) + 9 = 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 - 6y can be rewritten as (y - 3)^2 - 9.
Step 5: Substitute the completed squares back into the equation: ((x - 2)^2 - 4) + ((y - 3)^2 - 9) + 9 = 0.
Step 6: Simplify the equation: (x - 2)^2 + (y - 3)^2 - 4 = 0.
Step 7: Rearrange to isolate the squared terms: (x - 2)^2 + (y - 3)^2 = 4.
Step 8: Notice that the equation (x - 2)^2 + (y - 3)^2 = 0 represents a single point (2, 3).
Step 9: Since the equation represents a single point, the lines are coincident.

Conic Sections – Understanding the representation of conic sections, specifically circles and points, from quadratic equations.
Factoring and Completing the Square – The ability to rearrange and manipulate quadratic equations to identify their geometric representations.
Coincident Lines – Recognizing that coincident lines can be represented by a single point in the context of quadratic equations.