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Playbook

Conic Sections

44 q - 1.08/paper - 27% HARD. The chapter the 2025 syllabus added: 2 questions in the 29 shifts before 2025, then 15 in 2025 and 27 in 2026. Two notes pages, the parabola (18 q, 28% HARD) and the ellipse with the hyperbola (26 q, 27%), and the tangent conditions run through both. Nobody prepping from 2023-24 papers has seen it set.

Questions in the bank
44
q/paper (2025–26 shifts)
1.08
Tagged HARD
27%
Subtopics
2

Strand: Long Tail

When you’ll see it

A second-degree curve that is not a circle: y² = 4ax and its turns, an ellipse or hyperbola in x²/a² ± y²/b² = 1 form, an eccentricity, a focus, a directrix or a latus rectum, or a line asked to touch one of them.

How this chapter is tested

44 q, 1.08/paper, 27% HARD, and 42 of the 44 are from 2025 and 2026. This is the chapter the 2025 syllabus added: 2 questions in the 29 shifts before 2025, then 15 in the 13 shifts of 2025 and 27 in the 26 shifts of 2026, about one a paper both years. It crossed the playbook line on the 2026-10-13 re-measure, and nobody prepping from 2023-24 papers has seen it set.

Two notes pages (/notes/mht-cet-maths/conic-sections) at the same cost: the parabola (18 q, 28% HARD) and the ellipse with the hyperbola (26 q, 27%). Neither page is the cheap one, so the chapter does not cherry-pick; it is a short list of standard results that has to be learnt as a set.

Tangents run through both pages: 18 of the 44 stems name one. The condition is the same idea in three costumes, c = a/m for y² = 4ax, c² = a²m² + b² for an ellipse and c² = a²m² − b² for a hyperbola, and it answers a tangent of given slope, the pair of tangents from a point, a common tangent and two curves that cut at right angles. The rest is reading the parts of a curve: eccentricity (9 stems), a focus or the foci (8) and the latus rectum (6).

With no negative marking, the eccentricity questions are the cheapest elimination on the page: an ellipse's e is below 1 and a hyperbola's above it, so half the options usually go before any algebra.

The sub-skills

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

  • Parabola — standard forms, focus, directrix and latus rectum

    y² = 4ax has focus (a, 0), directrix x = −a and latus rectum 4a; the other three standard parabolas turn it. A shifted parabola is brought to standard form by completing the square, and the sign of the x or y coefficient says which way it opens.

  • Parabola — tangents

    y = mx + c touches y² = 4ax when c = a/m and x² = 4ay when c = −am². From a point (h, k) that becomes hm² − km + a = 0, whose two roots are the two tangent slopes; a common tangent to two parabolas satisfies both conditions.

  • Ellipse and hyperbola — eccentricity and foci

    Ellipse: b² = a²(1 − e²), foci (±ae, 0), e < 1. Hyperbola: b² = a²(e² − 1), e > 1. When the larger denominator sits under y² the major axis is vertical and the roles swap. A parametric curve is turned into x²/a² ± y²/b² = 1 before anything is read off it.

  • Ellipse and hyperbola — tangents and orthogonal curves

    y = mx ± √(a²m² + b²) for an ellipse and y = mx ± √(a²m² − b²) for a hyperbola, with the equation put in standard form first. An ellipse and a hyperbola with the same foci cut at right angles, the shortcut the orthogonal-curve stems are built on.

Traps to expect

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

  • The ellipse sign used for a hyperbola

    The ellipse has b² = a²(1 − e²) and c² = a²m² + b²; the hyperbola has b² = a²(e² − 1) and c² = a²m² − b². An option computed with the ellipse form is always present, and for a hyperbola's eccentricity it lands below 1, which rules it out on sight.

  • Reading 4a as a

    In x² = 20y, 4a = 20 and a = 5. Taking a = 20 puts the focus, the directrix and the latus rectum in the wrong places, and each of those wrong values is an option.

  • Directrix on the focus side

    The directrix is a from the vertex on the side AWAY from the focus. After completing the square on a parabola that opens left, the directrix is to the right of the vertex, and the mirrored line is the distractor.

  • Tangent condition applied before standard form

    9x² + 16y² = 288 is x²/32 + y²/18 = 1. Using 9 and 16 in c² = a²m² + b² gives a clean, wrong intercept that sits in the option list.

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.

Conic Sections notes

Drill every conic sections question

44 questions from the bank, scoped to 2 bundled subtopics.

Drill the 44 questions

Drill one subtopic at a time

The 2 subtopics this playbook covers, in catalog order.

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