Proving the Tangency of a Line to a Circle Using Different Methods

The problem at hand is to prove that the line yx is a tangent to the circle defined by the equation x^2y^2 - 4x^2 0. This can be achieved through several approaches, including using the concept of the discriminant and geometric properties of tangents and circles. Let's explore these methods in detail.

Using the Discriminant

The first method involves finding the point of intersection of the given line and the circle by substituting y x into the circle's equation.
Start by substituting y x into the circle's equation:
x2y2-4x20→x4-4x20 Next, simplify the equation:
x2(x2-4)0 This leads to a quadratic equation in terms of x^2. Furthermore, to find the discriminant of the equation x^4 - 4x^2 0, set it in the form b^2 - 4ac (-4)^2 - 4(1)(0) 16. Since the discriminant is zero, the quadratic equation has a single root, indicating that the line y x is tangent to the circle.

Geometric Approach

To verify the tangency, we can determine whether the line y x is perpendicular to a radius of the circle at a point on the circle. We start by completing the square for the circle's equation:

x2y2-4x2mo>mo-222 Thus, the center of the circle can be read off as (2, 0). The perpendicular family to the line yx is given by the equation xy constant, and since it passes through the center (2, 0), the constant must be 2. Therefore, the perpendicular from the center to the line y x intersects the line at (1, 1).

Now, let's verify that the point (1, 1) lies on the circle:

1212-412momo0 Since the point (1, 1) satisfies the circle's equation, the line yx is indeed tangent to the circle at this point.

Another Verification Method

A third approach is to use the formula for the length of the perpendicular from the center of the circle to the tangent line. The equation of the line is yx, and the circle's equation is x^2y^2 - 4x^2 0. Substituting yx into the circle's equation gives us:

x4-4x2momo0morightarrowmo2x^2 - 2x 1 0 This simplifies to a quadratic equation, and solving for x gives us the single solution x 1. Substituting x 1 back into the line equation yx gives us the point of tangency (1, 1).

To verify the tangency, we calculate the length of the perpendicular from the center (2, 0) to the line x - y 0. The perpendicular distance is:

2-01 1momofrac{2}{sqrt{2}}momosqrt{2} Since this length is equal to the radius of the circle, which is √2 units, the line is tangent to the circle at the point (1, 1).

Conclusion

Through these various methods, it has been shown that the line y x is indeed tangent to the circle defined by the equation x^2y^2 - 4x^2 0. The use of the discriminant, geometric properties, and perpendicular distance calculations all affirm the tangency point. This solution is crucial for understanding the relationship between lines and circles in coordinate geometry.