ARCH-MM-05: Circle Coordinate Geometry, Tangent Discriminants & Extremal Distances

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-05: Why the 170s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~170s (Blows 90s Clock)

Completing the square on

x^{2} - 4 x + y^{2} + 2 y - 4 = 0

gives

(x - 2)^2 + (y + 1)^2 = 9

, so the centre is

C(2, -1)

and radius is

r = 3

. By Pythagoras,

|CP| = \sqrt{(5 - (2))^2 + (3 + 1)^2} = \sqrt{3^2 + 4^2} = 5

. Subtracting the radius gives

|CP| - r = 5 - 3 = 2

(~140s).

The Fast ShortcutTarget: 45s (Saves 125s)

Line 1 (Halve & Negate Centre): Centre

C = (2, -1)

r = 3

; displacement

\vec{CP} = (3, 4) \implies |CP| = 5

(Pythagorean triple).
Line 2 (Subtract Radius):

d_{\min} = |CP| - r = 5 - 3 = 2

(30s).

Where Students Lose Time: In 23 past UAT-UK questions (5.61% of Mathematics 2), candidates lose 125s by forgetting to negate half the linear coefficients when completing the square. Instead, apply MM3 with Maths 1 skill M4.14.

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