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Conic Sections

The largest chapter on the paper, and it has grown. The circle, parabola, ellipse and hyperbola each come with their tangents; one tangency condition per curve answers most of them.

Questions in the bank
346
q/paper in 2025–26
2.97
Numeric answer
30%
Notes pages
10

Tier: Cornerstone

When you’ll see it

A circle, parabola, ellipse or hyperbola given by its equation, a focus or an eccentricity, or a line asked to touch one.

How this chapter is tested

The chapter runs circle first — its equation, its chords and tangents, then two circles together — and then gives the parabola, the ellipse and the hyperbola two pages each: the curve itself, then its tangents and normals. A last page joins two different curves.

Most questions are two steps: read the curve's numbers from its equation (centre and radius; a, b and e; the a of y² = 4ax), then compute what is asked. Marks slip in completing the square and in dividing out the x² coefficient, not in the formulas.

Time goes on tangents. One condition answers most of them: distance from the centre = radius for a circle, c = a/m for y² = 4ax, c² = a²m² + b² for an ellipse, c² = a²m² − b² for a hyperbola. Common-tangent questions apply two of these to one line. Focal chords and loci lean on the parametric point (at², 2at) and on straight lines.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • The equation of a circle

    Make the x² and y² coefficients 1, read the centre (−g, −f) and radius √(g² + f² − c), and build a circle from points, tangents or intercepts.

  • Chords and tangents of a circle

    Compare the distance from the centre with r; T = S₁ gives the chord with a given midpoint, √S₁ the tangent length from a point.

  • Two circles and families

    Compare the distance between centres with r₁ + r₂ and |r₁ − r₂|; S₁ − S₂ = 0 is the common chord, S₁ + λS₂ = 0 the circles through the meeting points.

  • The parabola and its tangents

    Write points as (at², 2at); t₁t₂ = −1 for a focal chord; tangent y = mx + a/m; normal y = mx − 2am − am³.

  • The ellipse and its tangents

    b² = a²(1 − e²), latus rectum 2b²/a, SP + S′P = 2a; tangent condition c² = a²m² + b²; director circle x² + y² = a² + b².

  • The hyperbola and its tangents

    b² = a²(e² − 1), |SP − S′P| = 2a, tangent condition c² = a²m² − b²; often paired with an ellipse through shared foci.

  • Common tangents and two curves

    One line, two tangency conditions with the same slope; the angle between curves from their slopes at the meeting point.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Reading g and f too early

    In 2x² + 2y² − 8x + 12y + 6 = 0 the centre is (2, −3), not (4, −6). Divide by the x² coefficient before reading anything.

  • Ellipse and hyperbola rules swapped

    b² = a²(1 − e²) with e < 1 is the ellipse; b² = a²(e² − 1) with e > 1 is the hyperbola. The tangency conditions differ by the sign of b² the same way, and the swapped value is usually an option.

  • −1 for the focus, −4 for the vertex

    On y² = 4ax a focal chord has t₁t₂ = −1, while a chord that subtends a right angle at the vertex has t₁t₂ = −4.

  • Features of the unshifted curve

    For (y − k)² = 4a(x − h) the focus is (h + a, k) and the directrix x = h − a. Shift a moved ellipse or hyperbola to the origin before using a tangency condition.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Conic Sections notes

Drill every Conic Sections question

346 questions from the bank, across 10 subtopics.

Drill the 346 questions

Drill one subtopic at a time

The 10 subtopics, in teaching order.

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