Playbook
Circle
47 q - 1.04/paper - 38% HARD. Tangent, Locus and Equation Construction is 27 of its 47 q. Two Circles (tangency, common tangents) is only 9 q but 56% HARD, the chapter's expensive corner.
- Questions in the bank
- 47
- q/paper (2024–25 shifts)
- 1.04
- Tagged HARD
- 38%
- Subtopics
- 3
Strand: Long Tail
When you’ll see it
A second-degree equation with equal coefficients on x squared and y squared, a tangency condition, or a distance measured from a point to a circle.
How this chapter is tested
47 q, 1.04/paper, 38% HARD. Two-thirds of it is ordinary coordinate geometry at ordinary cost: Tangent, Locus, and Equation Construction is 27 q at 37%, and Equation of Circle from Diameter, Centre, and Concentric Conditions is 11 q at 27% — the second-softest subtopic in the whole long tail. The expensive corner is Two Circles — Tangency, Common Tangents, and Relative Position, only 9 q but 56% HARD.
The chapter's most reusable move is not calculus. The greatest and least distance from an external point to a circle is the distance to the centre plus or minus the radius, full stop. The identical move answers 'greatest and least modulus of z on a disc' in Complex Numbers and 'maximum perpendicular distance from a point on a circle' here. At 1.8 minutes a question, replacing a calculus optimisation with one distance computation is a time lever, not merely an elegance.
Everything else is centre-and-radius bookkeeping. Read the centre as (-g, -f) and the radius as the square root of g squared plus f squared minus c, then compare a distance against that radius: less than means inside, equal means tangent, greater means outside. That one comparison drives point position, line position and the two-circle classification alike.
The two-circle corner is worth learning as a table rather than as a derivation: compare the distance between the centres against the sum and the absolute difference of the radii, and the number of common tangents (0, 1, 2, 3 or 4) follows from which case you are in.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Centre and radius from the general equation
For x^2 + y^2 + 2gx + 2fy + c = 0 the centre is (-g, -f) and the radius is the square root of (g^2 + f^2 - c). If that quantity is negative there is no real circle — a question that engineers this is testing whether you checked.
Constructing the equation
From centre and radius; from the two endpoints of a diameter using the diameter form; and from a concentric condition, where only the constant term changes.
Position of a point and of a line
Substitute the point into the left-hand side and read the sign; for a line, compare the perpendicular distance from the centre against the radius.
Tangent, normal and length of tangent
Condition of tangency is distance-from-centre equals radius. The length of the tangent from an external point is the square root of the left-hand side evaluated at that point. The normal always passes through the centre.
Two circles
Compare the distance between centres d against r1 + r2 and |r1 - r2|. Externally tangent when d = r1 + r2, internally tangent when d = |r1 - r2|, and the common-tangent count follows.
Geometric extremum without calculus
Greatest distance from an external point equals distance to centre plus radius; least equals distance to centre minus radius. Same move as the Complex Numbers modulus-on-a-disc family.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Radius left unsquare-rooted
g^2 + f^2 - c is the radius SQUARED. The option quoting it directly as the radius is standard, and it looks right.
Centre sign
The centre is (-g, -f), not (g, f). The sign-flipped centre, and every answer derived from it, is on the option list.
Unnormalised coefficients
If the coefficients of x^2 and y^2 are not 1, divide the whole equation through first. Reading g and f off the un-normalised form corrupts both centre and radius.
Tangent count from the wrong case
Touching internally gives 1 common tangent and touching externally gives 3; intersecting gives 2 and separated gives 4. Confusing the two tangency cases is the commonest error in the 56%-HARD corner.
Drill every circle question
47 questions from the bank, scoped to 3 bundled subtopics.
Drill one subtopic at a time
The 3 subtopics this playbook covers, in catalog order.
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.