The condition for the lines represented by the equation x^2 + y^2 - 4x - 6y + 9

The condition for the lines represented by the equation x^2 + y^2 - 4x - 6y + 9 = 0 to be coincident is:

The condition for the lines represented by the equation x^2 + y^2 - 4x - 6y + 9 = 0 to be coincident is:

Step 1: Start with the given equation: x^2 + y^2 - 4x - 6y + 9 = 0.
Step 2: Rearrange the equation to group x and y terms: (x^2 - 4x) + (y^2 - 6y) + 9 = 0.
Step 3: Complete the square for the x terms: x^2 - 4x becomes (x - 2)^2 - 4.
Step 4: Complete the square for the y terms: y^2 - 6y becomes (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 get the standard form of a circle: (x - 2)^2 + (y - 3)^2 = 4.
Step 8: Recognize that for the lines to be coincident, the discriminant of the quadratic must equal zero, which means the circle must touch itself at one point.

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