The lines represented by the equation x^2 - 6x + y^2 - 8y + 9 = 0 are:

The lines represented by the equation x^2 - 6x + y^2 - 8y + 9 = 0 are:

Step 1: Start with the equation x^2 - 6x + y^2 - 8y + 9 = 0.
Step 2: Rearrange the equation to group x and y terms: (x^2 - 6x) + (y^2 - 8y) + 9 = 0.
Step 3: Complete the square for the x terms: x^2 - 6x becomes (x - 3)^2 - 9.
Step 4: Complete the square for the y terms: y^2 - 8y becomes (y - 4)^2 - 16.
Step 5: Substitute the completed squares back into the equation: (x - 3)^2 - 9 + (y - 4)^2 - 16 + 9 = 0.
Step 6: Simplify the equation: (x - 3)^2 + (y - 4)^2 - 16 = 0.
Step 7: Rearrange to find the standard form of a circle: (x - 3)^2 + (y - 4)^2 = 16.
Step 8: Recognize that this represents a circle with center (3, 4) and radius 4.
Step 9: Understand that the lines represented by the equation are the tangents to the circle, which intersect at two distinct points.

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