Playbook
Circle
43 q - 1.00/paper - 33% HARD. Five notes pages. Tangents (13 q, 46% HARD) and Two Circles (7 q, 43%) are the expensive corners; the equation page is the cheap entry and the distance page is the greatest-and-least move shared with Complex Numbers.
- Questions in the bank
- 43
- q/paper (2024–25 shifts)
- 1.00
- Tagged HARD
- 33%
- Subtopics
- 5
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
43 q, 1.00/paper, 33% HARD, across five notes pages (/notes/mht-cet-maths/circle). Half of it is ordinary coordinate geometry at ordinary cost: the equation page is 12 q at 33% and concentric-and-touching 6 q at 17%. Tangents (13 q, 46% HARD) and Two Circles (7 q, 43%) are the expensive corners.
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 the maximum distance of a point of the circle from a line.
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 the line cuts, equal means tangent, greater means it misses. The tangent length from an external point is the square root of the circle's expression at that point, and the kite it makes with the two radii has area r times that length.
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.
Equation of a Circle — Centre-Radius, General, Diameter and Parametric Forms
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); the diameter form from two endpoints; the parametric form; and points of a family on a circle via Vieta. 12 q at 33% HARD.
Concentric Circles and Circles Touching a Line or an Axis
Same centre, new radius from an area or a point; radius equals the distance from the centre to the tangent line; the contact point is the foot of the perpendicular. 6 q at 17% HARD.
Tangents — At a Point, With a Given Slope, From an External Point and Their Loci
T = 0 at a point, y = mx ± a√(1 + m²) for a slope, the tangent length √S1 and the kite area, the angle between the tangents, and loci from tangent lengths. 13 q at 46% HARD — the chapter's expensive corner.
Distance From a Point to a Circle — Greatest, Least, a Line Cutting the Circle and the Segment Area
d ± r for the extreme distances, distance from the centre against r for a line, and the segment as a sector minus a triangle. 5 q at 0% HARD; the same move as the Complex Numbers modulus-on-a-disc family.
Two Circles — Touching, Common Tangents and Relative Position
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. 7 q at 43% HARD.
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.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.
Circle notesDrill every circle question
43 questions from the bank, scoped to 5 bundled subtopics.
Drill one subtopic at a time
The 5 subtopics this playbook covers, in catalog order.
- Equation of a Circle — Centre-Radius, General, Diameter and Parametric FormsDrill
- Concentric Circles and Circles Touching a Line or an AxisDrill
- Tangents — At a Point, With a Given Slope, From an External Point and Their LociDrill
- Distance From a Point to a Circle — Greatest, Least, a Line Cutting the Circle and the Segment AreaDrill
- Two Circles — Touching, Common Tangents and Relative PositionDrill
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.