Testing for tangents is a process used to determine if a line is tangent to a curve (usually a circle, ellipse, parabola, or more complex functions). A line is tangent to a curve at a particular point if it just touches the curve at that point without crossing it.

Here are some common methods used to test for tangency:

1. Algebraic Approach

• For circles, the line is tangent if the distance from the center of the circle to the line equals the circle’s radius.
• For more general functions $y = f(x)$, if a line has an equation $y = mx + b$, then to test if it’s tangent to $y = f(x)$, you set $f(x) = mx + b$ and check if the equation has exactly one solution (a single intersection point). This is because tangency implies they touch at only one point.

2. Using Derivatives (for differentiable curves)

• For curves defined by differentiable functions, you can compare the slope of the curve at a point to the slope of the line.
• If the slope of the tangent line (the derivative $f'(x)$) at a point on the curve matches the slope of the line, they may be tangent. You’d then check if they intersect at that point.

3. Implicit Differentiation (for implicit curves)

• For curves defined by an implicit equation (e.g., circles or ellipses given by ( x^