The figure shows a fixed circle C1 with equation...

The figure shows a fixed circle $C_{1}$ with equation $(x-1)^{2}+y^{2}=1$ and a shrinking circle $C_{2}$ with radius $r$ and center the origin. $P$ is the point $(0, r), Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line PQ and the $x$ -axis. What happens to Ras $C_{2}$ shrinks, that is, as $r \rightarrow 0^{+} ?$

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Key Concepts

Intersection of Circles

This concept involves finding the points where two circles meet by solving their equations simultaneously. It is a fundamental element of analytic geometry that allows one to determine critical points of contact between curves, which may be used to extract further geometric or algebraic information about a configuration.

Circle Equation

A circle is typically described by its equation in the form (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Understanding this equation is essential in analyzing how changes in the circle’s parameters affect its position and size in the coordinate plane.

Limit Process in Geometry

This refers to the investigation of how a geometric configuration changes as one or more of its parameters approach a particular value. It merges ideas from calculus and geometry, helping to determine the end behavior of figures—such as the eventual position of an intersection point—when a circle shrinks or expands.

Coordinate Geometry of Lines

Determining the equation of a line from given points and finding where it intersects an axis is a key component of coordinate geometry. This process involves calculating slopes and using point-slope or intercept forms of linear equations, which is vital for understanding the relationships and intersections among geometric figures.

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