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UAT-UK SPEC: MM3MODULE: MATHEMATICS 2TOPIC ID: ARCH-MM-05

Circle Coordinate Geometry, Tangent Discriminants & Extremal Distances

Master Circle Coordinate Geometry, Tangent Discriminants & Extremal Distances on the ESAT Mathematics 2 paper. Traditional classroom derivations waste valuable seconds. Using the 2-step Sub-90 invariant elimination method compresses solve time to 45s, saving +125s per question.

Standard Classroom Time
170s
Textbook multi-line derivations bleed exam clock
Sub-90 Target Pace
45s
Comfortably below the 88.8s per-question limit
Time Saved for Review
+125s
Banks clock buffer across 27 questions

2-Step Sub-90 Shortcut Formula

How to eliminate intermediate algebra and solve under 45 seconds.

Launch Interactive Lesson
STEP 1 • SETUP INVARIANT (LINE 1)10–15s
Line 1 (Halve & Negate Centre): Centre C=(2,−1)C = (2, -1), r=3r = 3; displacement CP⃗=(3,4)  ⟹  ∣CP∣=5\vec{CP} = (3, 4) \implies |CP| = 5 (Pythagorean triple).

Establish the structural invariant and eliminate impossible options immediately.

STEP 2 • FAST SOLVE (LINE 2)15–25s
Line 2 (Subtract Radius): dmin⁡=∣CP∣−r=5−3=2d_{\min} = |CP| - r = 5 - 3 = 2 (30s).

Evaluate the simplified expression using integer factoring or unit elimination.

Distractor Trap Forensics

The predictable cognitive traps engineered by examiners into the wrong options.

TRAP #1sign_or_direction_flip

Reading centre of x² + 2gx + y² + 2fy + c = 0 as (g, f) instead of (-g, -f)

Forgetting to negate half the linear coefficients when completing the square.

Sub-90 Fix: Test invariants before expanding equations.
TRAP #2factor_of_two_or_stoichiometry_trap

Using r = g² + f² - c instead of √(g² + f² - c)

Confusing r² with r.

Sub-90 Fix: Test invariants before expanding equations.
TRAP #3intermediate_stop_trap

Outputting centre-to-centre distance d instead of shortest boundary distance d - r₁ - r₂

Stopping at the distance between centres without subtracting the circle radii.

Sub-90 Fix: Test invariants before expanding equations.

Try 1 Timed Question: Circle Coordinate Geometry, Tangent Discriminants & Extremal Distances

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Full Lesson All 15 Questions (ARCH-MM-05)
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ARCH-MM-05: Circle Coordinate Geometry, Tangent Discriminants & Extremal Distances (Sample)Mathematics 2
00:00.0
SUB-90 ZONE (<45S)
0.0s45s Sub-9090s ESAT Budget150s Classroom
Question 1 of 1ARCH-MM-05 (6-Card Lesson)Format: Standard 5-Option
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Question 1 of 1 • 90s Target

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