If a and b are fixed numbers, find parametric equations for...

If $a$ and $b$ are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point $P$ in the figure, using the angle $\theta$ as the parameter. The line segment $A B$ is tangent to the larger circle.
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Key Concepts

Locus of Points

The concept of a locus in geometry refers to the set of all points that satisfy a given condition or set of constraints. In problems involving moving points under specific geometric restrictions, such as tangency conditions, the locus represents all the possible positions the point can occupy. Parametric equations are often used to express such loci, encapsulating the movement and constraints into a coherent mathematical description.

Parametric Equations

Parametric equations provide a way to describe a curve by expressing its coordinates as functions of one or more parameters. In the context of geometry, this method allows one to capture the entire path or locus of a moving point in the plane by varying the parameter, which is especially useful when dealing with curves that have a geometric construction or constraint.

Tangent Lines

A tangent line to a curve, and particularly to a circle, is a line that touches the curve at exactly one point and does not intersect it at that locality. The fundamental property of a tangent to a circle is that at the point of contact, the tangent is perpendicular to the radius. This relationship is crucial in linking the position of the point of tangency to the overall construction of the curve.

Trigonometric Parameterization

Using trigonometric functions like sine and cosine to parameterize curves is a common technique, especially when dealing with circles or portions of circular motion. By associating the coordinates of a point with trigonometric functions of an angle, one can elegantly represent circular arcs and related geometric curves, making it easier to manipulate and understand the relationships between different parts of a figure.

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