Coordinate geometry of lines and circles involves representing lines with slope-intercept, point-slope, or standard equations and circles with center-radius or expanded equations, interpreting parameters as geometric features. Problems are solved by using slope, distance, midpoint, and substitution or elimination to determine intersections, parallel or perpendicular relationships, tangency, and unknown coordinates, while distinguishing zero, one, and two intersection points. The scope is limited to Cartesian lines and circles with algebraic equations, excluding parametric or vector methods, general conic sections, and calculus-based analysis.
To find where a line and a circle intersect, use substitution:
Find the points where the line intersects the circle
The circle has center and radius . Since the line is , substitute for in the circle equation:
Simplify:
Taking the square root gives two possible values:
The line equation tells us that for both points. Therefore, the intersection points are
There are two intersection points because the line passes through the circle at two locations. In general, solving the substituted equation tells you the number of intersections: two real solutions mean two points, one real solution means the line is tangent to the circle, and no real solutions mean the line does not meet the circle.
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