The Architecture of a Locus
A Locus is not a static shape; it is the path mapped out by a moving point $P(x, y)$ that obeys a strict mathematical rule. By converting this rule into an algebraic equation, the invisible path becomes visible.
The Equidistant Rule
Rule: Point $P$ must stay exactly the same distance from Point $A$ as it is from Point $B$.
The Result: The moving point will always form a perfectly straight line (the perpendicular bisector of AB).
The Ratio Rule
Rule: Point $P$ must stay exactly $k$ times further from Point $A$ than from Point $B$.
The Result: If $k \neq 1$, the algebraic expansion will always form an Apollonius Circle.
Guided Proof: The Equidistant Locus
Two beacons are located at A(0, 0) and B(4, 8). A moving receiver $P(x,y)$ is programmed to always stay perfectly equidistant ($PA = PB$) between them. Prove the equation of its path algebraically.
Step 1: Set up the Squared Distances
Use Pythagoras. $PA^2 = x^2 + y^2$. Expand $PB^2 = (x-4)^2 + (y-8)^2$. What is the final constant term of $PB^2$ after expanding $16 + 64$?
Step 2: Equate and Cancel
Equate: $x^2 + y^2 = x^2 - 8x + y^2 - 16y + 80$. The squares cancel out! Move all variables to the left. $8x + \dots y = 80$. What is the coefficient of $y$?
Step 3: Equation of the Path
Rearrange $8x + 16y = 80$ into $y = mx + c$ form to reveal the linear path.
Parameters: The Apollonius Ratio
Given fixed points $A$ and $B$, changing the distance ratio multiplier ($k$) of the moving point $P$ fundamentally alters the geometry of the locus.
Rule: $PA = PB$. The point $P$ must stay equidistant from $A(-2,1)$ and $B(2,5)$.
The Linear Path
When you expand $PA^2 = PB^2$ using these asymmetrical coordinates, the $x^2$ and $y^2$ terms completely cancel out! The result is a perfect straight line (the perpendicular bisector of AB). Here, it simplifies to $y = -x + 3$.
Rule: $PA = 2PB \Rightarrow PA^2 = 4PB^2$. The point must stay exactly twice as far from A(-3,0) as it is from B(3,0).
Step 1: The Multiplication Matrix
Set up: $(x+3)^2+y^2 = 4((x-3)^2+y^2)$. Expand LHS: $x^2+6x+9+y^2$. Expand RHS: $4(x^2-6x+9+y^2) = 4x^2 - 24x + 36 + \dots y^2$. What is the coefficient of $y^2$ on the RHS?
Step 2: Simplify the Circle
Move everything to RHS: $3x^2 - 30x + 3y^2 + 27 = 0$. Divide entirely by 3 to get standard form: $x^2 + \dots x + y^2 = -9$. What is the coefficient of $x$?
Step 3: Complete the Square
Complete the square for $x^2 - 10x$: $(x - 5)^2 - 25$. So $(x - 5)^2 + y^2 = -9 + 25 = 16$. What is the radius of this Apollonius Circle?
Rule: $2PA = PB \Rightarrow 4PA^2 = PB^2$. The point must stay exactly half the distance from A(-3,0) as it is from B(3,0).
Step 1: The Multiplication Matrix
Set up: $4((x+3)^2+y^2) = (x-3)^2+y^2$. Expand LHS: $4x^2 + 24x + 36 + 4y^2$. Expand RHS: $x^2 - 6x + \dots + y^2$. What is the constant term on RHS?
Step 2: Simplify the Circle
Move everything to LHS: $3x^2 + 30x + 3y^2 + 27 = 0$. Divide entirely by 3: $x^2 + \dots x + y^2 = -9$. What is the coefficient of $x$?
Step 3: Complete the Square
Complete the square for $x^2 + 10x$: $(x + 5)^2 - 25$. So $(x + 5)^2 + y^2 = 16$. The circle mirrors the previous ratio! What is the x-coordinate of the center?
Scholarship: Threat / Safe Zones
The Aggro Radius
The Game Engine Setup
In a stealth game, an AI Guard is at Tower T(12, 0). The Player's spawn point is at Base E(0,0). The engine calculates a "Safe Zone" locus where the player is mathematically guaranteed to be at least twice as far from the Guard as they are from the Base.
The Mathematics
The rule $d_T = 2 \times d_E$ translates to $PT^2 = 4PE^2$. Expanding this generates a circular boundary. Inside the boundary, the player is safe. Outside, the Guard aggros.
Calculate the exact equation of the Safe Zone boundary if Base is E(0,0), Tower is T(12,0), and the locus rule is $d_T = 2d_E$.
Step 1: Set up the Squared Rule
Equation: $(x-12)^2 + y^2 = 4(x^2 + y^2)$. Expand the LHS: $x^2 - 24x + \dots + y^2$. What is the constant term?
Step 2: Balance the Equation
Move everything to RHS to keep $x^2$ positive: $3x^2 + 24x + 3y^2 = 144$. Divide by 3: $x^2 + \dots x + y^2 = 48$. Coefficient of x?
Step 3: Define Safe Zone Boundary
Complete the square: $(x+4)^2 - 16 + y^2 = 48 \Rightarrow (x+4)^2 + y^2 = 64$. What is the precise radius of the Safe Zone?