Conic Sections

Class 11 · Mathematics

4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.

Circles

Part of: Circle and Parabola

Fixed-distance points form circles; centre-radius form squares gaps; completing squares reveals hidden centres; diameter endpoints average to centres.

Why it matters. Circles model wheels and orbits. Centre-radius form plots instantly.

Formulas

(x - h)^2 + (y - k)^2 = r^2
Centre (-g, -f), radius = sqrt(g^2 + f^2 - c)
Diameter ends (x1, y1) and (x2, y2): centre is the midpoint

Key words

Circle: : The set of points at a fixed distance from a fixed centre.
Centre-radius form: : The equation (x - h)^2 + (y - k)^2 = r^2 for centre (h, k).
General form: : The equation x^2 + y^2 + 2gx + 2fy + c = 0 with centre (-g, -f).
Diameter form: : The fact that a diameter's endpoints average to the centre.

Try it in two minutes

Fix centre and radius.
Square the gaps.
Complete the squares.

Try at home. Draw a circle with string compass.

Parabolas

Part of: Circle and Parabola

Equal distance from focus and directrix defines parabolas; y-squared-four-ax opens right with focus (a, 0); latus rectum spans four-a through focus; axis mirrors the curve.

Why it matters. Parabolas focus signals. Standard forms aim dishes.

Formulas

y^2 = 4ax opens right, focus (a, 0), directrix x + a = 0
x^2 = 4ay opens upward, focus (0, a), directrix y + a = 0
Latus rectum = 4a

Key words

Parabola: : The set of points at equal distance from a focus and a directrix.
Focus: : The fixed point whose distance defines the parabola, as (a, 0) for y^2 = 4ax.
Directrix: : The fixed line whose distance defines the parabola, as x + a = 0 for y^2 = 4ax.
Latus rectum: : The chord through the focus across the axis, of length 4a.
Axis: : The mirror line of the parabola through its focus and vertex.

Try it in two minutes

Spot the squared variable.
Read a from coefficient.
Mark focus and directrix.

Try at home. Trace y = x² dots.

Ellipses

Part of: Ellipse and Hyperbola

Constant focal sum shapes ellipses; bigger denominator anchors major axis; c-squared equals a-squared-minus-b-squared; eccentricity under one; latus rectum two-b-squared-over-a.

Why it matters. Orbits are ellipses. Eccentricity measures roundness.

Formulas

x^2 / a^2 + y^2 / b^2 = 1
c^2 = a^2 - b^2
e = c / a with 0 <= e <= 1
Latus rectum = 2b^2 / a

Key words

Ellipse: : The set of points whose distances from two foci add to a constant.
Major axis: : The longer axis of the ellipse, anchored by the bigger denominator.
Minor axis: : The shorter axis of the ellipse, across the major axis.
Eccentricity: : The ratio e = c / a, below 1 for every ellipse.
Latus rectum: : The chord through a focus across the major axis, of length 2b^2 / a.

Try it in two minutes

Find the bigger denominator.
Compute c from gap.
Read e as c-over-a.

Try at home. Draw ellipse with two pins.

Hyperbolas

Part of: Ellipse and Hyperbola

Constant focal gap shapes hyperbolas; plus-sign variable anchors transverse axis; c-squared sums a-squared-b-squared; eccentricity exceeds one; asymptotes cross with slopes plus-minus-b-over-a.

Why it matters. Hyperbolas aim navigation signals. Gaps define branches.

Formulas

x^2 / a^2 - y^2 / b^2 = 1
c^2 = a^2 + b^2
e = c / a with e >= 1
Asymptotes: y = (b / a)x and y = -(b / a)x

Key words

Hyperbola: : The set of points whose distances from two foci differ by a constant.
Transverse axis: : The axis holding the vertices and foci, anchored by the plus-sign variable.
Conjugate axis: : The axis across the transverse axis through the centre.
Asymptotes: : The crossing lines the hyperbola approaches, with slopes plus or minus b / a.
Eccentricity: : The ratio e = c / a, above 1 for every hyperbola.

Try it in two minutes

Find the plus variable.
Sum squares for c.
Read e above one.

Try at home. Sketch y = 1/x branches.

What your child will practise

Circles: Candy Puzzle (Clue Puzzle)
Circles: Candy Quick Fire (Quick Fire)
Ellipses: Desert Puzzle (Clue Puzzle)
Hyperbolas: Space Quest (Flip and Match)
Parabolas: Ocean Quick Fire (Quick Fire)

Quiz: Circles (8 questions)
Quiz: Ellipses (8 questions)
Quiz: Hyperbolas (8 questions)
Quiz: Parabolas (8 questions)

Conic Sections

Class 11 · Mathematics

4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.

Circles

Part of: Circle and Parabola

Fixed-distance points form circles; centre-radius form squares gaps; completing squares reveals hidden centres; diameter endpoints average to centres.

Why it matters. Circles model wheels and orbits. Centre-radius form plots instantly.

Formulas

(x - h)^2 + (y - k)^2 = r^2
Centre (-g, -f), radius = sqrt(g^2 + f^2 - c)
Diameter ends (x1, y1) and (x2, y2): centre is the midpoint

Key words

Circle: : The set of points at a fixed distance from a fixed centre.
Centre-radius form: : The equation (x - h)^2 + (y - k)^2 = r^2 for centre (h, k).
General form: : The equation x^2 + y^2 + 2gx + 2fy + c = 0 with centre (-g, -f).
Diameter form: : The fact that a diameter's endpoints average to the centre.

Try it in two minutes

Fix centre and radius.
Square the gaps.
Complete the squares.

Try at home. Draw a circle with string compass.

Parabolas

Part of: Circle and Parabola

Equal distance from focus and directrix defines parabolas; y-squared-four-ax opens right with focus (a, 0); latus rectum spans four-a through focus; axis mirrors the curve.

Why it matters. Parabolas focus signals. Standard forms aim dishes.

Formulas

y^2 = 4ax opens right, focus (a, 0), directrix x + a = 0
x^2 = 4ay opens upward, focus (0, a), directrix y + a = 0
Latus rectum = 4a

Key words

Parabola: : The set of points at equal distance from a focus and a directrix.
Focus: : The fixed point whose distance defines the parabola, as (a, 0) for y^2 = 4ax.
Directrix: : The fixed line whose distance defines the parabola, as x + a = 0 for y^2 = 4ax.
Latus rectum: : The chord through the focus across the axis, of length 4a.
Axis: : The mirror line of the parabola through its focus and vertex.

Try it in two minutes

Spot the squared variable.
Read a from coefficient.
Mark focus and directrix.

Try at home. Trace y = x² dots.

Ellipses

Part of: Ellipse and Hyperbola

Constant focal sum shapes ellipses; bigger denominator anchors major axis; c-squared equals a-squared-minus-b-squared; eccentricity under one; latus rectum two-b-squared-over-a.

Why it matters. Orbits are ellipses. Eccentricity measures roundness.

Formulas

x^2 / a^2 + y^2 / b^2 = 1
c^2 = a^2 - b^2
e = c / a with 0 <= e <= 1
Latus rectum = 2b^2 / a

Key words

Ellipse: : The set of points whose distances from two foci add to a constant.
Major axis: : The longer axis of the ellipse, anchored by the bigger denominator.
Minor axis: : The shorter axis of the ellipse, across the major axis.
Eccentricity: : The ratio e = c / a, below 1 for every ellipse.
Latus rectum: : The chord through a focus across the major axis, of length 2b^2 / a.

Try it in two minutes

Find the bigger denominator.
Compute c from gap.
Read e as c-over-a.

Try at home. Draw ellipse with two pins.

Hyperbolas

Part of: Ellipse and Hyperbola

Constant focal gap shapes hyperbolas; plus-sign variable anchors transverse axis; c-squared sums a-squared-b-squared; eccentricity exceeds one; asymptotes cross with slopes plus-minus-b-over-a.

Why it matters. Hyperbolas aim navigation signals. Gaps define branches.

Formulas

x^2 / a^2 - y^2 / b^2 = 1
c^2 = a^2 + b^2
e = c / a with e >= 1
Asymptotes: y = (b / a)x and y = -(b / a)x

Key words

Hyperbola: : The set of points whose distances from two foci differ by a constant.
Transverse axis: : The axis holding the vertices and foci, anchored by the plus-sign variable.
Conjugate axis: : The axis across the transverse axis through the centre.
Asymptotes: : The crossing lines the hyperbola approaches, with slopes plus or minus b / a.
Eccentricity: : The ratio e = c / a, above 1 for every hyperbola.

Try it in two minutes

Find the plus variable.
Sum squares for c.
Read e above one.

Try at home. Sketch y = 1/x branches.

What your child will practise

Circles: Candy Puzzle (Clue Puzzle)
Circles: Candy Quick Fire (Quick Fire)
Ellipses: Desert Puzzle (Clue Puzzle)
Hyperbolas: Space Quest (Flip and Match)
Parabolas: Ocean Quick Fire (Quick Fire)

Quiz: Circles (8 questions)
Quiz: Ellipses (8 questions)
Quiz: Hyperbolas (8 questions)
Quiz: Parabolas (8 questions)