Shunya / Circles & Conic Sections

Shunya · Maths lab

Circles & conic sections

Drag points round a circle to test the theorems about chords, angles and tangents, work out arcs, sectors and segments, see a line cut a circle, then meet the parabola, the ellipse and the hyperbola with their foci, directrices and eccentricity, and watch one slider turn a circle into all of them.

Lessons

Pick your class, follow the steps in the lab, and check what you see against the working under the drawing.

Class 9 — Chords and the centre

Concept: the perpendicular from the centre of a circle to a chord bisects the chord. Equal chords are equally far from the centre, and a longer chord is nearer the centre.

Open Circle theorems and choose Chords. Drag A and B to move the chord AB. The dashed line from O meets AB at its midpoint at a right angle.
Read the two cards: the length of AB and its distance d from O. Check $AB = 2\sqrt{r^2 - d^2}$ with your own numbers.
Drag the second chord CD until the two lengths match. The distances match too.
Pull a chord through the centre: d = 0 and the chord is a diameter, the longest chord of all.

Takeaway: the closer a chord is to the centre, the longer it is, and the perpendicular from the centre always cuts it in half.

Class 9 — Angle at the centre and angles in a segment

Concept: the arc AB subtends an angle at the centre that is twice the angle it subtends at any point P on the rest of the circle. Angles in the same segment are equal.

Open Circle theorems and choose Angle at the centre. Drag P round the circle: ∠APB never changes and the arc AB is always exactly twice it.
Press P on the minor arc. Now the arc that does not contain P is the major arc, so the angle at the centre is reflex: still twice ∠APB.
Choose Angles in a segment. With P and Q on the same side of AB, ∠APB = ∠AQB, however you move them.
Press opposite segments: the two angles now add up to 180°.

Takeaway: every angle standing on the same arc is half the angle at the centre, so they are all equal.

Class 9 — Angle in a semicircle and cyclic quadrilaterals

Concept: the angle in a semicircle is a right angle. The opposite angles of a cyclic quadrilateral add up to 180°, and an exterior angle equals the interior opposite angle.

Choose Angle in a semicircle. AB is a diameter. Drag P anywhere: ∠APB stays 90°, and PA² + PB² = AB².
Choose Cyclic quadrilateral and drag A, B, C and D round the circle. The cards ∠A + ∠C and ∠B + ∠D both read 180°.
Look at the exterior angle at C: it equals the interior angle A.

Takeaway: a quadrilateral with all four corners on a circle has opposite angles that are supplementary, and the converse is also true.

Class 10 — Tangents to a circle

Concept: a tangent touches the circle at one point and is perpendicular to the radius there. The two tangents from an outside point are equal in length: $PT^2 = OP^2 - r^2$ .

Choose Tangents and drag P outside the circle. The two tangents meet the circle at T₁ and T₂, and the right-angle marks show OT ⟂ PT.
Check PT₁ = PT₂ = √(OP² − r²) on the cards. Move P closer: the tangents get shorter, and when P is on the circle there is only one.
Drag P inside the circle: no tangent exists.
Look at ∠T₁PT₂ + ∠T₁OT₂ = 180°. Then try Intersecting chords: for any point P, PA · PA′ = PB · PB′.

Takeaway: the radius to the point of contact is perpendicular to the tangent, and tangents from one point are equal.

Class 10 — Areas related to circles

Concept: an arc, a sector and a segment are the same fraction $\tfrac{\theta}{360^\circ}$ of the circle, and the segment is the sector minus the triangle.

Open Arcs & sectors. Drag B round the circle to change the central angle θ.
Press 90°: the sector is a quarter of the circle, and the arc length is a quarter of the circumference.
Tick shade the segment and show the triangle OAB: segment = sector − triangle.
Press 180° then 270°: the same formulas work for a major sector, and the minor and major segments add up to the whole circle.

Takeaway: arc $= \tfrac{\theta}{360^\circ} \times 2\pi r$ and sector $= \tfrac{\theta}{360^\circ} \times \pi r^2$ .

Class 11 — Equation of a circle, and a line meeting it

Concept: the circle with centre (h, k) and radius r has the equation $(x - h)^2 + (y - k)^2 = r^2$ . A line is a tangent when its distance from the centre equals the radius.

Open Circle & line. Read the standard and the general form of the circle and drag the centre: the equation follows.
Drag the two dots on the line. The card tells you if it is a secant (d < r), a tangent (d = r) or misses (d > r).
Press Make it a tangent and check the tangent condition $(c + mh - k)^2 = r^2(1 + m^2)$ in the working.
Switch to Two circles and press each preset: see how d compared with r₁ + r₂ and |r₁ − r₂| decides the number of common tangents (4, 3, 2, 1 or 0).

Takeaway: compare the distance d with the radius to classify a line, and compare d with r₁ + r₂ and |r₁ − r₂| to classify two circles.

Class 11 — The parabola

Concept: a parabola is every point that is as far from the focus as from the directrix. With the vertex at the origin, $y^2 = 4ax$ has its focus at (a, 0) and directrix x = −a.

Open Parabola and press y² = 8x. Here a = 2, so the focus is (2, 0) and the directrix is x = −2.
Drag P along the curve and watch PF and PM: they are always equal.
Drag the focus F away from the vertex: a grows and the parabola widens. Tick latus rectum to see its length 4a.
Turn on tangent at P and the reflection property: rays from the focus leave parallel to the axis.
Try the four directions and the shifted parabolas such as (y − 1)² = 8(x − 2).

Takeaway: the parabola is defined by focus and directrix, its latus rectum is 4a, and its eccentricity is exactly 1.

Class 11 — The ellipse

Concept: an ellipse is every point whose distances from two foci add up to a constant, $PF_1 + PF_2 = 2a$ . With $c^2 = a^2 - b^2$ , the eccentricity is $e = \tfrac{c}{a} < 1$ .

Open Ellipse and press x²/25 + y²/9 = 1. Read a = 5, b = 3, c = 4, e = 0.8 and the foci (±4, 0).
Drag P round the curve: PF₁ + PF₂ stays 2a = 10.
Drag the two axis dots. When the axes are equal the foci merge and the ellipse is a circle (e = 0). Make it thin: e gets close to 1.
Tick directrices and latus rectum and check $2b^2 \div a$ against the card.
Turn on reflection: a ray from one focus always passes through the other.

Takeaway: the foci are c from the centre, b² = a² − c², and the area is πab.

Class 11 — The hyperbola

Concept: a hyperbola is every point whose distances from two foci differ by a constant, $|PF_1 - PF_2| = 2a$ . Here $c^2 = a^2 + b^2$ and $e = \tfrac{c}{a} > 1$ .

Open Hyperbola and press x²/16 − y²/9 = 1. Read a = 4, b = 3, c = 5, e = 1.25 and the asymptotes y = ±¾x.
Drag P: |PF₁ − PF₂| stays 2a = 8. P can jump between the two branches.
Drag the conjugate-axis dot to change b and watch the asymptotes tilt: the curve always hugs them.
Press rectangular (a = b): the asymptotes are perpendicular and e = √2. Switch to opens up and down for y²/a² − x²/b² = 1.

Takeaway: compared with the ellipse the plus becomes a minus: c² = a² + b², and the asymptotes are the lines y = ±(b ÷ a)x.

Class 11 — Eccentricity: one family of curves

Concept: every conic is the set of points with $PF = e \times PM$ , where M is the foot of the perpendicular to the directrix. The eccentricity e decides which curve you get.

Open Eccentricity and press circle: e = 0, then ellipse: e = 0.5. The curve stretches as e grows.
Press parabola: e = 1: the curve just opens up and escapes to infinity.
Press hyperbola: e = 1.5: now there are two branches (blue is the second).
Drag P and read PF ÷ PM: it always equals e.
Move the slider slowly from 0 to 3 and watch the four families appear in turn.

Takeaway: e = 0 circle, 0 < e < 1 ellipse, e = 1 parabola, e > 1 hyperbola: they are all the same curve seen at different eccentricities.

The maths behind it

What each mode covers

Circle theorems — seven theorems on one circle you can drag: the angle at the centre, angles in a segment, the angle in a semicircle, cyclic quadrilaterals, chords, tangents and intersecting chords. Class 9 and 10 circles.

Arcs & sectors — drag the central angle for the arc length, the sector area and the segment area, with the working. Class 10 areas related to circles.

Circle & line — the equation of a circle in standard and general form, a line that is a secant, a tangent or outside, the tangent condition, and two circles with their common tangents. Class 10 and 11.

Parabola — focus, directrix, latus rectum, the PF = PM definition, tangents and the reflection property, in all four directions. Class 11 conic sections.

Ellipse — foci, eccentricity, directrices, latus rectum, area, PF₁ + PF₂ = 2a, tangents and the reflection property. Class 11 conic sections.

Hyperbola — foci, asymptotes and the guide rectangle, eccentricity, directrices, |PF₁ − PF₂| = 2a and the rectangular hyperbola. Class 11 conic sections.

Eccentricity — one slider from e = 0 to 3 that turns a circle into an ellipse, a parabola and a hyperbola, with PF ÷ PM = e checked at any point. Class 11 conic sections.

Part of Shunya. See the syllabus map for the chapters these modes map to. The Geometry lab has the compass constructions, including tangents from a point. Also try the Function & Graph Explorer.

Formulas used in this lab

Chords Class 9

Length of a chord at distance d from the centre: $AB = 2\sqrt{r^2 - d^2}$
Perpendicular from the centre: $OM \perp AB \ \Rightarrow\ AM = MB$
Equal chords: $AB = CD \iff OM = ON$
Longer chord, nearer the centre: $AB > CD \iff OM < ON$
The diameter is the longest chord: $d = 0 \ \Rightarrow\ AB = 2r$

Angles in a circle Class 9

Angle at the centre: $\angle AOB = 2\,\angle APB$
Same segment: $\angle APB = \angle AQB$
Opposite segments: $\angle APB + \angle AQB = 180^\circ$
Angle in a semicircle: $\angle APB = 90^\circ$
Converse (P then lies on the circle with diameter AB): $\angle APB = 90^\circ$

Cyclic quadrilaterals Class 9

Opposite angles: $\angle A + \angle C = 180^\circ,\quad \angle B + \angle D = 180^\circ$
Exterior angle: $\text{exterior angle at } C = \angle A$
Converse: $\angle A + \angle C = 180^\circ \Rightarrow ABCD \text{ is cyclic}$
Angle sum: $\angle A + \angle B + \angle C + \angle D = 360^\circ$

Tangents Class 10

Tangent and radius: $OT \perp PT$
Tangent length: $PT^2 = OP^2 - r^2$
Two tangents from a point: $PT_1 = PT_2$
Angle between the tangents: $\angle T_1PT_2 + \angle T_1OT_2 = 180^\circ$
Intersecting chords: $PA \cdot PA' = PB \cdot PB'$
Power of a point: $PA \cdot PA' = |OP^2 - r^2|$

Areas related to circles Class 10

Circumference and area: $C = 2\pi r,\qquad A = \pi r^2$
Arc length: $l = \dfrac{\theta}{360^\circ} \times 2\pi r$
Area of a sector: $\dfrac{\theta}{360^\circ} \times \pi r^2$
Area of a segment: $\text{sector} - \text{triangle} = \dfrac{r^2}{2}\,(\theta - \sin\theta)$
Chord of angle θ: $AB = 2r\sin\dfrac{\theta}{2}$
Perimeter of a sector: $l + 2r$

Circle and line Class 11

Standard form: $(x - h)^2 + (y - k)^2 = r^2$
General form: $x^2 + y^2 + 2gx + 2fy + c = 0$
Centre and radius: $(-g,\ -f),\qquad r = \sqrt{g^2 + f^2 - c}$
Distance from the centre to y = mx + c: $d = \dfrac{|mh - k + c|}{\sqrt{1 + m^2}}$
Secant, tangent, outside: $d < r,\quad d = r,\quad d > r$
Tangent condition: $(c + mh - k)^2 = r^2(1 + m^2)$
Two circles: $d > r_1 + r_2,\quad d = r_1 + r_2,\quad |r_1 - r_2| < d < r_1 + r_2,\ \dots$

Parabola Class 11

Standard forms: $y^2 = 4ax,\quad y^2 = -4ax,\quad x^2 = 4ay,\quad x^2 = -4ay$
Focus and directrix of y² = 4ax: $F = (a,\ 0),\qquad x = -a$
Definition: $PF = PM$
Latus rectum: $4a$
Eccentricity: $e = 1$
Tangent at (x₁, y₁): $y\,y_1 = 2a\,(x + x_1)$

Ellipse Class 11

Standard form: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1,\qquad a > b$
Distance to the foci: $c^2 = a^2 - b^2$
Eccentricity: $e = \dfrac{c}{a} < 1$
Foci and directrices: $(\pm ae,\ 0),\qquad x = \pm\dfrac{a}{e}$
Definition: $PF_1 + PF_2 = 2a$
Latus rectum: $\dfrac{2b^2}{a}$
Area: $\pi a b$

Hyperbola Class 11

Standard form: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$
Distance to the foci: $c^2 = a^2 + b^2$
Eccentricity: $e = \dfrac{c}{a} > 1$
Foci and directrices: $(\pm ae,\ 0),\qquad x = \pm\dfrac{a}{e}$
Definition: $|PF_1 - PF_2| = 2a$
Asymptotes: $y = \pm\dfrac{b}{a}\,x$
Latus rectum: $\dfrac{2b^2}{a}$

Eccentricity and the family Class 11

Focus–directrix definition: $PF = e \cdot PM$
Polar form (focus at the origin): $r = \dfrac{\ell}{1 + e\cos\theta}$
Kind of conic: $e = 0:\ \text{circle},\quad e < 1:\ \text{ellipse},\quad e = 1:\ \text{parabola},\quad e > 1:\ \text{hyperbola}$
Semi-major axis of an ellipse: $a = \dfrac{\ell}{1 - e^2}$
Semi-transverse axis of a hyperbola: $a = \dfrac{\ell}{e^2 - 1}$
Relation between a, b, e: $b^2 = a^2(1 - e^2)\ \text{(ellipse)},\qquad b^2 = a^2(e^2 - 1)\ \text{(hyperbola)}$