Conic sections
Slice a carrot straight across and you get a circle; slice it at a slant and you get an oval. Slicing a cone with a plane gives all the curves in this chapter.
A conic section is the curve in which a plane cuts a (double) cone. If the plane is perpendicular to the axis you get a circle; slightly tilted, an ellipse; parallel to a slant line of the cone, a parabola; cutting both halves, a hyperbola. Special cases of a plane through the vertex give a point, a straight line or a pair of lines.
Rule 1: Name the conic — Think about the angle of the cutting plane.
- Easy: Plane perpendicular to the axis: circle.
- Medium: Plane parallel to a slant side: parabola.
- Exam-type: Plane parallel to the axis, cutting both nappes: hyperbola.
Rule 2: Real-life conics — Look for these curves around you.
- Easy: A football-shaped path of a planet around the Sun: ellipse (nearly circular for Earth).
- Medium: A thrown ball's path and a satellite dish cross-section: parabola.
- Exam-type: Cooling-tower sides: hyperbola.
Every conic can be described by a focus and a fixed ratio e (eccentricity): e = 0 circle, e < 1 ellipse, e = 1 parabola, e > 1 hyperbola.
Thinking an ellipse and a circle are unrelated; a circle is a special ellipse.
Circle, Ellipse, Parabola, Hyperbola: tilt the plane more and more.
Spot the difference
| Curve | Cutting plane |
|---|---|
| Circle | perpendicular to the axis |
| Ellipse | slightly tilted |
| Parabola | parallel to a slant side |
| Hyperbola | steeper, cuts both halves |
Practice questions
- 1Which conic is obtained when a plane is perpendicular to the axis of a cone?
- 2Which conic is obtained when a plane is parallel to a slant side of the cone?
- 3Which conic cuts both nappes of the cone?
- 4Name the curve obtained when a plane passes through the vertex of the cone and touches only the vertex.
- 5What is the eccentricity of a circle?
- 6Which conics have eccentricity less than 1?
- 7A plane through the vertex and along a slant line gives what degenerate conic?
- 8A plane through the vertex cutting both nappes gives what?
Show answers (try them yourself first)
- 1Circle
- 2Parabola
- 3Hyperbola
- 4A point
- 50
- 6Circle and ellipse
- 7A straight line
- 8A pair of intersecting straight lines
Equation of a circle
A compass draws all points at the same distance from the pin. That is exactly the circle.
A circle is the set of all points at a fixed distance r (radius) from a fixed point C(h, k) (centre). Its equation is (x − h)² + (y − k)² = r²; with centre at the origin it is x² + y² = r².
Formula: (x − h)² + (y − k)² = r²
In the figure: The circle x² + y² = 9 has centre (0, 0) and radius 3.
Rule 1: Write the equation — Put h, k and r in the formula.
- Easy: Centre (0, 0), radius 5: x² + y² = 25.
- Medium: Centre (2, 3), radius 4: (x − 2)² + (y − 3)² = 16.
- Exam-type: Centre (−1, 2), radius 3: (x + 1)² + (y − 2)² = 9.
Rule 2: Read centre and radius — Compare with the standard form.
- Easy: (x − 4)² + (y + 1)² = 36: centre (4, −1), radius 6.
- Medium: x² + y² = 49: centre (0, 0), radius 7.
- Exam-type: (x + 3)² + y² = 2: centre (−3, 0), radius √2.
Signs inside the brackets are the opposite of the centre coordinates.
Writing r instead of r² on the right-hand side.
Minus the centre, squared; r squared on the right.
Practice questions
- 1Write the circle with centre (0, 0) and radius 6.
- 2Write the circle with centre (1, −2) and radius 3.
- 3Find the centre and radius of (x + 2)² + (y − 5)² = 25.
- 4A circle has centre (3, 4) and passes through the origin. Find its equation.
- 5A circle has centre (2, 1) and passes through (5, 5). Find its radius.
- 6Find the circle with diameter endpoints (0, 0) and (6, 8).
- 7Does (3, 4) lie on x² + y² = 25?
- 8Is (4, 4) inside x² + y² = 25?
Show answers (try them yourself first)
- 1x² + y² = 36
- 2(x − 1)² + (y + 2)² = 9
- 3Centre (−2, 5), radius 5
- 4r² = 9 + 16 = 25: (x − 3)² + (y − 4)² = 25
- 5√(9 + 16) = 5
- 6centre (3, 4), r = 5: (x − 3)² + (y − 4)² = 25
- 79 + 16 = 25, yes
- 816 + 16 = 32 > 25, so outside
General form of a circle
Sometimes the equation comes expanded. Completing the square brings back the centre and radius.
Expanding gives x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f) and radius √(g² + f² − c). To read an equation, complete the square in x and in y.
Formula: x² + y² + 2gx + 2fy + c = 0
Rule 1: Find centre and radius — Complete the squares.
- Easy: x² + y² − 4x + 6y − 12 = 0: (x − 2)² + (y + 3)² = 25; centre (2, −3), r = 5.
- Medium: x² + y² + 2x − 8 = 0: (x + 1)² + y² = 9; centre (−1, 0), r = 3.
- Exam-type: x² + y² − 6x − 8y = 0: (x − 3)² + (y − 4)² = 25; passes through the origin.
Rule 2: Check if it is a circle — g² + f² − c must be positive.
- Easy: x² + y² + 4x + 6y + 13 = 0: 4 + 9 − 13 = 0, a single point (−2, −3).
- Medium: x² + y² + 2x + 2y + 5 = 0: 1 + 1 − 5 < 0, no real circle.
- Exam-type: x² + y² − 2x − 3 = 0: 1 + 0 + 3 = 4, radius 2.
The coefficients of x² and y² must be equal and there must be no xy term.
Forgetting to take half the coefficient before squaring.
Half, square, add on both sides.
Practice questions
- 1Find the centre of x² + y² − 6x + 4y − 3 = 0.
- 2Find the radius of x² + y² − 6x + 4y − 3 = 0.
- 3Find the centre and radius of x² + y² + 8x = 0.
- 4Find the radius of x² + y² − 2x + 2y − 7 = 0.
- 5Find the equation of the circle with centre (1, 1) and radius 2 in general form.
- 6For which k is x² + y² + 2x + k = 0 a real circle (positive radius)?
- 7Find the equation of the circle through (0, 0), (4, 0) and (0, 6).
- 8Find the centre of the circle x² + y² − 10x − 4y + 4 = 0.
Show answers (try them yourself first)
- 1(3, −2)
- 2√(9 + 4 + 3) = 4
- 3Centre (−4, 0), radius 4
- 4√(1 + 1 + 7) = 3
- 5x² + y² − 2x − 2y − 2 = 0
- 6g = 1: r² = 1 − k > 0, so k < 1
- 7x² + y² − 4x − 6y = 0
- 8(5, 2), radius 5
Circles from conditions
To draw a circle you need its centre and one point on it, or three points on it.
Use the known data: (i) centre and a point → r = distance; (ii) ends of a diameter → centre is the midpoint; (iii) three points → solve for g, f, c.
Rule 1: Centre and a point — Radius is the distance from the centre to the point.
- Easy: Centre (0, 0), through (3, 4): r = 5.
- Medium: Centre (1, 2), through (4, 6): r = 5.
- Exam-type: Centre (−2, 1), through (1, 5): r = 5.
Rule 2: Tangent to axes — A circle touching the x-axis has r = |k|; touching both axes has |h| = |k| = r.
- Easy: Centre (3, 3) touching both axes: radius 3.
- Medium: Centre (0, 4) touching the x-axis: radius 4.
- Exam-type: Centre (5, −5) touching both axes: radius 5.
Distance between two points: √((x₂ − x₁)² + (y₂ − y₁)²).
Using the radius where the diameter is required.
Centre + point → distance → r.
Practice questions
- 1Find r for a circle with centre (0, 0) through (6, 8).
- 2Find the circle with centre (2, 0) touching the y-axis.
- 3Find the circle with centre (3, 3) touching both axes.
- 4Find the circle with diameter ends (2, 3) and (6, 7).
- 5Find the radius of the circle with centre (1, 1) through (4, 5).
- 6Find the circle with centre (−1, −1) through the origin.
- 7The circle (x − 1)² + (y − 2)² = 9 and the line x = 4: how many common points?
- 8Find the equation of the circle with centre on the x-axis through (0, 0) and (6, 0).
Show answers (try them yourself first)
- 110
- 2(x − 2)² + y² = 4
- 3(x − 3)² + (y − 3)² = 9
- 4centre (4, 5), r² = 8: (x − 4)² + (y − 5)² = 8
- 55
- 6(x + 1)² + (y + 1)² = 2
- 7Distance of centre from line = 3 = radius: one point (tangent)
- 8centre (3, 0), r = 3: (x − 3)² + y² = 9