What is the angle between the lines represented by the equation x^2 - 6x + y^2 -

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Question: What is the angle between the lines represented by the equation x^2 - 6x + y^2 - 8y + 9 = 0?

Options:

Correct Answer: 90 degrees

Solution:

By completing the square, we can find the slopes of the lines and calculate the angle between them.

What is the angle between the lines represented by the equation x^2 - 6x + y^2 - 8y + 9 = 0?

What is the angle between the lines represented by the equation x^2 - 6x + y^2 - 8y + 9 = 0?

Step 1: Start with the given 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 the equation: (x - 3)^2 + (y - 4)^2 = 16.
Step 8: Recognize that this is the equation of a circle with center (3, 4) and radius 4.
Step 9: Since the equation represents a circle, it does not have lines with slopes to find an angle between.
Step 10: Conclude that the angle between the lines represented by the equation is not applicable.

Completing the Square – This technique is used to rewrite quadratic equations in a form that reveals their geometric properties, such as the slopes of lines.
Finding the Angle Between Lines – This involves using the slopes of the lines derived from the quadratic equation to calculate the angle using the tangent formula.