Schooling · Class 11 · Mathematics · Conic Sections · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Conic Sections: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- A conic section is the locus of a…
- For a circle, e = 0; for an ellipse…
- A circle is the set of all points…
- A parabola has one focus and one…
- An ellipse has two foci; its…
- A hyperbola has two foci and two…
- The latus rectum is the chord…
- Formulas and facts
- Circle
- Parabola
- Ellipse (a > b)
- Hyperbola
- For an ellipse, the sum of distances…
- For a hyperbola, the absolute…
- Asymptotes of the hyperbola x²/a² −…
- Worked examples
- Finding the equation of a parabola
- Equation of an ellipse from given data
- Identifying a hyperbola and its…
- Watch out for
- Confusing c² = a² − b² (ellipse)…
- Using diameter instead of radius…
- Forgetting that for an ellipse a is…
- Plotting a parabola y² = 4ax as if…
- Mixing up latus rectum formulas
What this chapter is about
Conic sections are curves obtained when a plane cuts through a double-napped right circular cone at different angles. Depending on the angle and position of the cutting plane, you get a circle, an ellipse, a parabola or a hyperbola. These curves appear everywhere in nature and engineering: the path of a thrown ball is a parabola, planets move in elliptical orbits, and satellite dish reflectors use parabolic shapes to focus signals.
Key ideas
- A conic section is the locus of a point whose distance from a fixed point (focus) bears a constant ratio (eccentricity, e) to its distance from a fixed line (directrix).
- For a circle, e = 0; for an ellipse, 0 < e < 1; for a parabola, e = 1; for a hyperbola, e > 1.
- A circle is the set of all points equidistant from a fixed centre; its equation with centre (h, k) and radius r is (x − h)² + (y − k)² = r².
- A parabola has one focus and one directrix; its simplest forms are y² = 4ax (opening rightward) and x² = 4ay (opening upward).
- An ellipse has two foci; its standard form with centre at origin is x²/a² + y²/b² = 1, where a > b > 0 and c² = a² − b² gives the distance from centre to each focus.
- A hyperbola has two foci and two branches; its standard form is x²/a² − y²/b² = 1, where c² = a² + b² and the asymptotes are y = ±(b/a)x.
- The latus rectum is the chord through a focus perpendicular to the principal axis; its length helps in sketching and in problems involving focal chords.
- Every conic can also be written in general second-degree form Ax² + Bxy + Cy² + Dx + Ey + F = 0; the value of B² − 4AC decides the type (circle, ellipse, parabola or hyperbola) when B = 0 reduces the analysis.
Formulas and facts to remember
- Conic: Circle · Standard equation: x² + y² = r² · Eccentricity: 0 · Foci: centre itself · Directrix: none · Latus rectum length: 2r (diameter)
- Conic: Parabola · Standard equation: y² = 4ax · Eccentricity: 1 · Foci: (a, 0) · Directrix: x = −a · Latus rectum length: 4a
- Conic: Ellipse (a > b) · Standard equation: x²/a² + y²/b² = 1 · Eccentricity: c/a, c² = a² − b² · Foci: (±c, 0) · Directrix: x = ±a/e · Latus rectum length: 2b²/a
- Conic: Hyperbola · Standard equation: x²/a² − y²/b² = 1 · Eccentricity: c/a, c² = a² + b² · Foci: (±c, 0) · Directrix: x = ±a/e · Latus rectum length: 2b²/a
- For an ellipse, the sum of distances from any point on the curve to the two foci equals 2a.
- For a hyperbola, the absolute difference of distances from any point on the curve to the two foci equals 2a.
- Asymptotes of the hyperbola x²/a² − y²/b² = 1 are y = (b/a)x and y = −(b/a)x.
Common mistakes
- Confusing c² = a² − b² (ellipse) with c² = a² + b² (hyperbola) → remember, ellipse has smaller c because e < 1.
- Using diameter instead of radius when writing a circle's equation → always check that you square the radius, not the diameter.
- Forgetting that for an ellipse a is always the larger denominator, so first rewrite the equation to identify a and b correctly.
- Plotting a parabola y² = 4ax as if it opens upward → it opens along the positive x-axis; x² = 4ay opens upward.
- Mixing up latus rectum formulas: 4a for parabola, 2b²/a for ellipse and hyperbola → keep a short list handy until they become automatic.
Quick revision
- Circle: all points equidistant from centre; e = 0.
- Parabola: e = 1; vertex, focus and directrix determine the curve; latus rectum = 4a.
- Ellipse: 0 < e < 1; sum of focal distances = 2a; c² = a² − b².
- Hyperbola: e > 1; difference of focal distances = 2a; c² = a² + b²; has two asymptotes.
- Standard equations assume centre (or vertex) at origin and axes along coordinate axes.