Exploring the Angles Between the Lines Represented by the Equation x2 - y2 0

The equation x^2 - y^2 0 is a prime example of a hyperbola, which can be factored into two linear equations. Understanding the relationship between these lines and the angle they form is crucial in various mathematical and practical applications.

Factoring the Equation

The equation x^2 - y^2 0 can be factored using the difference of squares formula:

x^2 - y^2 (x - y)(x y) 0

This results in the following two linear equations:

x - y 0 or y x x y 0 or y -x

Identifying the Lines and Their Slopes

Let's identify the lines and their slopes:

Line 1: y x with a slope of m_1 1 Line 2: y -x with a slope of m_2 -1

Calculating the Angle Between the Lines

To find the angle θ between these two lines, we can use the formula for the angle between two lines given their slopes:

tan θ left| frac{m_1 - m_2}{1 m_1 m_2} right|

Substituting the slopes into the formula:

tan θ left| frac{1 - (-1)}{1 (1) cdot (-1)} right| left| frac{1 1}{1 - 1} right| left| frac{2}{0} right|

Since the denominator becomes zero, this leads to an undefined expression. This indicates that the lines are perpendicular. Therefore, the angle between them is:

θ 90°

Alternative Method for Verifying Perpendicularity

Another method to verify the perpendicularity of the lines is to use the concept of the angle between two curves. If we have two curves with equations y_1(x) and y_2(x), the angle αx between those curves at each value of x is given by:

αx tan^{-1} left| frac{y_1'(x) - y_2'(x)}{1 y_1'(x) cdot y_2'(x)} right|

In this specific case, we have:

x^2 - y^2 0

This implies:

y^2 x^2 or y ±x

Setting y_1(x) x and y_2(x) -x, we get:

y_1'(x) 1 and y_2'(x) -1

Substituting these into the formula:

αx tan^{-1} left| frac{1 - (-1)}{1 1 cdot (-1)} right| tan^{-1} left| frac{2}{0} right| frac{π}{2}

This confirms that the lines are indeed perpendicular, and αx is independent of x, meaning the lines are perpendicular at any value of x, not just at the intersection point x 0.

Conclusion

In conclusion, the equation x^2 - y^2 0 represents two lines that are perpendicular to each other, forming an angle of 90 degrees. This understanding is valuable in various mathematical and practical scenarios, including geometry and calculus. The methods outlined above provide a robust way to determine the relationship between these lines and the angle they form.