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75-Archetype SyllabusMathematics 2 • ARCH-MM-05 • Spec MM3
Full 6-Card Lesson 15-Question Timed Drill
UAT-UK SPEC CODE: MM3 (Coordinate geometry)ASSUMED MATHS 1 PREREQUISITE: M4.14FREQUENCY: HIGH

Circle Coordinate Geometry, Tangent Discriminants & Extremal Distances

Deconstructed from 23 past UAT-UK questions (5.61% of all historical Mathematics 2 papers). Students using standard A-Level classroom methods spend an average of 170s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 45s (saving 125s per question).

Historical Frequency
23 Past Qs
5.61% of Mathematics 2
Classroom Trap Time
170s
+80s over 90s budget
Sub-90 Target Time
45s
Banks +125s on the clock
Dominant UI Format
5-Option A–E
Maths 1 Skill: M4.14

Card 1 & Card 2 Preview: The Classroom Time Trap vs. Sub-90 2-Line Rule

Excerpted from our interactive 6-Step Topic Lesson for ARCH-MM-05.

Launch All 6 Active-Recall Cards
CARD 1 OF 6 • THE CLASSROOM TIME TRAP (170s)
Why Students Bleed Time:
Completing the square on x2−4x+y2+2y−4=0x^{2} - 4 x + y^{2} + 2 y - 4 = 0 gives (x−2)2+(y+1)2=9(x - 2)^2 + (y + 1)^2 = 9, so the centre is C(2,−1)C(2, -1) and radius is r=3r = 3. By Pythagoras, ∣CP∣=(5−(2))2+(3+1)2=32+42=5|CP| = \sqrt{(5 - (2))^2 + (3 + 1)^2} = \sqrt{3^2 + 4^2} = 5. Subtracting the radius gives ∣CP∣−r=5−3=2|CP| - r = 5 - 3 = 2 (~140s).
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.
CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<45s)
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).
Line 2 (Subtract Radius): dmin⁡=∣CP∣−r=5−3=2d_{\min} = |CP| - r = 5 - 3 = 2 (30s).
Sanity & Limit Check: Verify units/signs against sign_or_direction_flip (Reading centre of x² + 2gx + y² + 2fy + c = 0 as (g, f) instead of (-g, -f)) and factor_of_two_or_stoichiometry_trap (Using r = g² + f² - c instead of sqrt(g² + f² - c)).

Examiner Distractor Blueprint for ARCH-MM-05

FOIL #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.

Open Sign / Direction Flip Clinic
FOIL #2factor_of_two_or_stoichiometry_trap
Using r = g² + f² - c instead of sqrt(g² + f² - c)

Confusing r² with r.

Open Factor-of-2 / Stoichiometry Clinic
FOIL #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.

Open Intermediate Stop Trap Clinic
FOIL #4fence_post_or_domain_trap
Setting discriminant b² - 4ac > 0 without checking if intersection points lie on the restricted segment

Ignoring domain bounds on the line segment or semi-circle.

Open Fence-Post / Domain Trap Clinic

Try 1 Live Sample Question Under the 90-Second Stopwatch

Press W to open the 2-line dry-wipe scratchpad, select A–E, and lock in to reveal the Dual-View solution.

Full 6-Card Lesson Launch All 15 Questions (ARCH-MM-05)
sub90 ESAT Stopwatch Emblemsub90 ESAT
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
Right-click any option to cross out • Keys A–E + Enter
Question 1 of 1 • 90s Target

Ready to Start Question 1?

The question stem, diagram, and answer options are hidden behind this pre-start gate. When you click below, the question will reveal and your 90-second split stopwatch will immediately begin ticking.

Mathematics 2•ARCH-MM-05•Standard 5-Option
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