I am given an equation..
x^2-y^2-2x-6y = 8
Move things around...
(x^2-2x)-(y^2-6y) = 8
I need to find out what kind of graph this makes. How do I do this?
To determine what kind of graph the equation represents, we can start by simplifying it and rearranging it into a standard form of a conic section.
1. Simplify the equation by grouping the x-terms and the y-terms:
(x^2 - 2x) - (y^2 - 6y) = 8
2. Complete the square for both the x-terms and the y-terms to express them as perfect squares:
(x^2 - 2x + 1) - 1 - (y^2 - 6y + 9) + 9 = 8
(x^2 - 2x + 1) - (y^2 - 6y + 9) = 8 + 1 - 9
(x - 1)^2 - (y - 3)^2 = 0
3. Rearrange the equation to isolate the squared terms on one side and constant terms on the other side:
(x - 1)^2 - (y - 3)^2 = 0
Now, let's analyze the resulting equation:
If the equation of a conic section contains only a squared term with a coefficient of 1 and no constant term, it represents a degenerate conic section, specifically a pair of intersecting lines.
In this case, the equation (x - 1)^2 - (y - 3)^2 = 0 represents a degenerate conic section where two lines intersect at point (1, 3).
So, the graph of this equation represents two intersecting lines passing through the point (1, 3).