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CBSE Class 11 · Mathematics

Conic Sections

Official NCERT chapter from Mathematics (book code kemh1). ExamMaster notes are original teaching at CBSE Class 11 depth.

This lesson follows the official chapter “Conic Sections” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 11
  • Medium level
  • 8 concepts

1Sections of a Cone

A conic is a slice of a cone: circle, ellipse, parabola, hyperbola — as the lesson named the cuts (parallel to the base, steeper, through the vertex-story). Section is the cut-shape, not a random oval doodle.

A freehand egg is not a named conic until the cut or the equation is argued.

Figure. A plane cut on a right circular cone is classified by eccentricity: e<1 ellipse (circle when e=0), e=1 parabola, e>1 hyperbola.

How it works

  1. Name the cone and the cutting plane as taughtThe section.
  2. Hang circle / ellipse / parabola / hyperbolaThe family.
  3. Refuse a look-only “roundish”A named cut.

2Circle

A circle is (x−h)²+(y−k)²=r²: all points at distance r from (h,k). Centre (2, −1), r=3 is (x−2)²+(y+1)²=9. A circle is equal radii, not a pretty O.

x²+y²+4x=0 needs a complete-square before you read centre and r.

Figure. Circle x^2+y^2=25 on centred axes. Centre is the origin; radius 5 is the run from O to (5,0).

How it works

  1. Name centre and rThe given.
  2. Write (x−h)²+(y−k)²=r²The circle.
  3. Complete the square if the writing is mixedThe read.

Centre (2, −1), r=3

Write the circle’s equation.

  • Form(x−2)²+(y+1)²=9
  • Expand if askedx²+y²−4x+2y−4=0
  • Read backcentre (2,−1), r=3

Pro tip. Equal distance r from the centre.

3Parabola

A parabola at this class is a focus-directrix story or y²=4ax (and the taught twins) as the standard write. Distance to focus equals distance to the line. A satellite-dish photo is a setting; 4a is the length that names the focus.

Calling every U-shape y=x² without a focus is a doodle.

Figure. Standard parabola y^2=12x: read 4a=12 so a=3. The focus sits at (3,0) on the axis. The vertex is the origin.

How it works

  1. Name the standard write or the focus and directrixThe given.
  2. Keep PF = (distance to directrix)The definition.
  3. Read a from 4a if that form was usedThe parabola.

4Ellipse

An ellipse is a two-focus story: PF1+PF2 = 2a (constant), or x²/a²+y²/b²=1 as taught. A stretched circle-look is not enough until the sum-or-the-write is named. For a>b, the major axis is the 2a story.

Using one focus as if it were a circle’s centre is the family-steal.

Figure. Ellipse x^2/25+y^2/9=1. Larger denominator is a^2=25, so e=4/5 and the foci sit at (\pm 4,0) on the major axis.

How it works

  1. Name the two foci or the standard writeThe given.
  2. Keep the constant sum 2aThe ellipse.
  3. Read a, b from the denominatorsThe axes.

5Hyperbola

A hyperbola is a two-focus difference: |PF1−PF2|=2a, or x²/a²−y²/b²=1 as taught. Two branches, not one oval. A difference is not a sum — that was the ellipse.

Drawing two facing parabolas by eye is not this test.

Figure. Hyperbola x^2/16-y^2/9=1 with dashed asymptotes y=\pm(3/4)x. Foci sit at (\pm 5,0); e=5/4>1.

How it works

  1. Name the difference 2a or the standard writeThe given.
  2. Keep two branchesThe hyperbola.
  3. Refuse a sum-of-distances as this headingDifference, not sum.

6A definition is a test you can run

Conic is a test: a named cut or a named focus-story plus the standard write. If you only say “curve”, you have a heading.

A logo-oval is not the test.

Figure. Parallel means a constant gap. A transversal makes matching angle pairs — those equals are facts you mark, not a tidy-looking sketch.

How it works

  1. Name circle / parabola / ellipse / hyperbola and the writeThe object.
  2. Give r, 4a, 2a-sum, or 2a-differenceThe test.
  3. Then the word has contentThe definition ran.

7Name the given before the unknown

The given is the centre/r or a, b. The unknown is the equation or the type. Copy the minus in x²/a²−y²/b² before you call it an ellipse.

Using r=9 because the writing had =9, when that 9 was r², is a silent given.

Figure. Name the given coefficient first. From y^2=12x you read 4a=12, then a=3, then the focus (3,0) — never start at the unknown.

How it works

  1. Copy the write or the focus-dataThe given.
  2. Name type or the expanded formThe unknown.
  3. Then complete the square or read a,bGiven first.

8One worked case is enough at this class

One standard write is enough. A page of the same r=3 clone does not add a new hyperbola.

Ten twins of (x−2)²+(y+1)²=9 are still one idea.

Figure. One standard-form case is enough at this class: the coefficient 12 gives a=3 and the focus (3,0). Do not start a second family until this reading is stable.

How it works

  1. Write one circle or one ellipseThe case.
  2. A second only as a checkOptional.
  3. StopThis class.
(x−2)²+(y+1)²=9 is a circle of radius
  1. 3
  2. 9
  3. 2

r²=9, so r=3.

Notes

  • The official chapter title is “Conic Sections”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • circle: (x−h)²+(y−k)²=r²
  • ellipse: PF1+PF2=2a
  • hyperbola: |PF1−PF2|=2a

Recap

Hold these pegs from the official chapter “Conic Sections”. The wording is ExamMaster’s teaching, not a textbook recap.

Sections of a Cone
A conic is a slice of a cone: circle, ellipse, parabola, hyperbola — as the lesson named the cuts (parallel to the base, steeper, through the vertex-story).
Circle
A circle is (x−h)²+(y−k)²=r²: all points at distance r from (h,k).
Parabola
A parabola at this class is a focus-directrix story or y²=4ax (and the taught twins) as the standard write.
Ellipse
An ellipse is a two-focus story: PF1+PF2 = 2a (constant), or x²/a²+y²/b²=1 as taught.
Hyperbola
A hyperbola is a two-focus difference: |PF1−PF2|=2a, or x²/a²−y²/b²=1 as taught.
A definition is a test you can run
Conic is a test: a named cut or a named focus-story plus the standard write.

Practise Conic Sections

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