The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 8 = 0 are:

₹0.0

Question: The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 8 = 0 are:

Options:

Correct Answer: Intersecting

Solution:

To determine the nature of the lines, we can rewrite the equation in the form of (x - a)^2 + (y - b)^2 = r^2 and analyze the discriminant.

The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 8 = 0 are:

The pair of lines represented by the equation x^2 - 4x + y^2 - 6y + 8 = 0 are:

Step 1: Start with the given equation: x^2 - 4x + y^2 - 6y + 8 = 0.
Step 2: Rearrange the equation to isolate the constant on one side: x^2 - 4x + y^2 - 6y = -8.
Step 3: Complete the square for the x terms (x^2 - 4x). To do this, take half of -4 (which is -2), square it (getting 4), and add it to both sides: (x - 2)^2 - 4.
Step 4: Complete the square for the y terms (y^2 - 6y). Take half of -6 (which is -3), square it (getting 9), and add it to both sides: (y - 3)^2 - 9.
Step 5: Substitute the completed squares back into the equation: (x - 2)^2 - 4 + (y - 3)^2 - 9 = -8.
Step 6: Simplify the equation: (x - 2)^2 + (y - 3)^2 - 13 = -8, which gives (x - 2)^2 + (y - 3)^2 = 5.
Step 7: Now, we have the equation in the form (x - a)^2 + (y - b)^2 = r^2, where a = 2, b = 3, and r^2 = 5.
Step 8: Since r^2 is positive (5 > 0), this indicates that the equation represents a circle, not a pair of lines.

Conic Sections – Understanding how to identify and analyze the equations of conic sections, particularly circles and lines.
Discriminant Analysis – Using the discriminant to determine the nature of the roots of a quadratic equation, which can indicate whether the conic represents intersecting lines, parallel lines, or a single line.
Completing the Square – Rewriting quadratic equations in a standard form to facilitate analysis of their geometric representation.