AI Tutor
An air traffic controller spots an unidentified drone cruising in a straight line at 450 miles per hour toward a restricted flight zone. How do you calculate the exact second it enters radar coverage, how long it stays visible, and its closest distance to the airport tower?
To solve problems like this, engineers map real-world ranges into Cartesian coordinates, turning physical boundaries into algebraic equations.

Circular Detection Zones

A circle is the set of all points at a fixed distance rr from a center (h,k)(h, k). By the Pythagorean theorem, any point (x,y)(x, y) on the boundary satisfies (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2.
When modeling broadcast or radar coverage, the detection zone includes the interior, described by the inequality (x−h)2+(y−k)2≤r2(x - h)^2 + (y - k)^2 \le r^2.
What happens when two broadcast stations overlap, and where can a receiver pick up both signals?

Dual-Station Coverage and Overlap

Two radio towers, Tower A at (0,0)(0, 0) with a 10-mile broadcast range and Tower B at (12,0)(12, 0) with an 8-mile range, create overlapping coverage.
The points receiving signals from both towers must satisfy the system of inequalities x2+y2≤100x^2 + y^2 \le 100 and (x−12)2+y2≤64(x - 12)^2 + y^2 \le 64.
To find the boundary line of this shared region, subtract the two circle equations: (x2+y2)−((x−12)2+y2)=100−64(x^2 + y^2) - ((x - 12)^2 + y^2) = 100 - 64 simplifies directly to 24x−144=3624x - 144 = 36, giving x=7.5x = 7.5.
Substituting x=7.5x = 7.5 into x2+y2=100x^2 + y^2 = 100 yields y2=100−56.25=43.75y^2 = 100 - 56.25 = 43.75, so the boundary intersection points are (7.5,43.75)≈(7.5,6.61)(7.5, \sqrt{43.75}) \approx (7.5, 6.61) and (7.5,−6.61)(7.5, -6.61).