Circle and Conic Sections

Chapter 10 · Coordinate Geometry · 3 questions in the paper · 11 min read

Circle and Conic Sections

What the syllabus asks for 2
  • Equation of a circle, centre and radius, tangent
  • Parabola, ellipse and hyperbola in standard form

Conics carry about three questions: the centre or radius of a circle, the focus or latus rectum of a parabola, and the eccentricity or axis of an ellipse or hyperbola. Everything is a substitution into a standard form; the work is recognising which form you have.

What the paper asks from this chapter
  • Circle: (x-h)^2+(y-k)^2=r^2, general form, tangent conditions.
  • Parabola y^2=4ax: focus, directrix, latus rectum.
  • Ellipse and hyperbola: axes, foci, eccentricity, asymptotes.
  • Eccentricity: circle 0, ellipse < 1, parabola 1, hyperbola > 1.

10.1 Circle: equation, centre, radius, tangent

  • Centre (h,k), radius r: (x-h)^2+(y-k)^2=r^2. Centre at the origin: x^2+y^2=r^2.
  • General form x^2+y^2+2gx+2fy+c=0: centre (-g,-f), radius \sqrt{g^2+f^2-c}. It passes through the origin when c=0.
  • A line touches a circle when the distance from the centre to the line equals r.
  • Tangent at (x_1,y_1) to x^2+y^2=r^2: xx_1+yy_1=r^2. Length of tangent from an outside point =\sqrt{x_1^2+y_1^2-r^2}.
  • A point is inside, on or outside according as x_1^2+y_1^2-r^2 is negative, zero or positive.

photograph
(a) Apollonius, Conica (printed edition)
photograph
(b) Kepler: planets move on ellipses
photograph
(c) Parabolic dish: all rays to the focus
Plate 10.1. The people behind the conics. (a) A page of Apollonius of Perga's Conica (about 200 BC), the first book to name the parabola, ellipse and hyperbola. (b) Johannes Kepler (1571-1630), who showed that planets move on ellipses with the Sun at one focus. (c) A parabolic dish: every incoming ray is sent to the single focus, which is why satellite dishes and torch reflectors are parabolas.Photo credits: (a) Giuseppe Cocchini, book by Apollonius of P, Public domain; (b) August Köhler [1], Public domain; (c) Adamantios, CC BY-SA 3.0.

diagram
Figure 10.1. A circle from its general equation, computed. x^2 + y^2 - 6x + 4y - 12 = 0 has g = -3, f = 2, so centre (3, -2) and radius sqrt(9 + 4 + 12) = 5. The point (6, 2) lies on it, and the tangent there is perpendicular to the radius: slope -3/4, equation 3x + 4y - 26 = 0, whose distance from the centre is exactly 5. From the outside point (3, 6) the tangent length is sqrt(64 - 25) = sqrt(39) = 6.24.

10.2 Parabola, ellipse and hyperbola in standard form

Conic Equation Key facts
Parabola y^2=4ax vertex (0,0), focus (a,0), directrix x=-a, latus rectum 4a, e=1
Parabola x^2=4ay opens upward, focus (0,a)
Ellipse (a>b) \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 e=\sqrt{1-\frac{b^2}{a^2}}, foci (\pm ae,0), major axis 2a, latus rectum \frac{2b^2}{a}, area \pi ab
Hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 e=\sqrt{1+\frac{b^2}{a^2}}, foci (\pm ae,0), asymptotes y=\pm\frac{b}{a}x
Rect. hyperbola xy=c^2 or x^2-y^2=a^2 asymptotes at right angles

For an ellipse the larger denominator belongs to the major axis, so \frac{x^2}{16}+\frac{y^2}{25}=1 has its major axis along the y-axis. Distance between the foci =2ae.

diagram
Figure 10.2. The three standard conics with their numbers computed. (a) Parabola y^2 = 8x: a = 2, focus (2, 0), directrix x = -2, latus rectum 4a = 8. (b) Ellipse x^2/25 + y^2/9 = 1: a = 5, b = 3, c = sqrt(25 - 9) = 4, e = 4/5 = 0.8, foci (+-4, 0). (c) Hyperbola x^2/16 - y^2/9 = 1: a = 4, b = 3, c = sqrt(16 + 9) = 5, e = 5/4 = 1.25, asymptotes y = +-(3/4)x.

diagram
Figure 10.3. Eccentricity classifies every conic. e = 0 is a circle, 0 < e < 1 an ellipse, e = 1 a parabola and e > 1 a hyperbola. The scale shows the conics computed above: the circle of radius 5 (e = 0), the ellipse (0.8), the parabola (1) and the hyperbola (1.25). Right: the same curves arise when a plane slices a cone at steeper and steeper angles.

10.3 Eight traps in circles and conics

Conic questions come from a small set of formulas, and the hardest part is knowing which letters mean what. These eight traps are drawn from the model questions on circles, parabolas, ellipses and hyperbolas.

  1. The centre in (x-h)^2+(y-k)^2=r^2 is (h,k); in the general form it is (-g,-f). For (x-3)^2+(y+2)^2=25 the centre is (3,-2), so the sign inside the bracket is opposite to the coordinate. For x^2+y^2-6x+4y-12=0 we have g=-3, f=2, centre (3,-2), radius \sqrt{9+4+12}=5 (Fig 10.1). The radius squared must be positive, or there is no real circle.
  2. Test a point by comparing distances. Compute S_1=x_1^2+y_1^2+2gx_1+2fy_1+c for the point: negative means inside, zero means on the circle and positive means outside. The tangent length from an outside point is \sqrt{S_1}. For (3,6) that is \sqrt{39}=6.24.
  3. A tangent is perpendicular to the radius. A line touches a circle when its distance from the centre equals r. The tangent at (x_1,y_1) on x^2+y^2=r^2 is xx_1+yy_1=r^2. If the radius slope is \frac43 the tangent slope is -\frac34.
  4. Use the diameter form when two end points are given. The circle with diameter ends (x_1,y_1) and (x_2,y_2) is (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0. This is faster than finding the centre and radius separately. A circle through the origin has c=0.
  5. Parabola y^2=4ax has focus (a,0), directrix x=-a and latus rectum 4a. For y^2=8x, a=2, so the focus is (2,0) (Fig 10.2). The form x^2=4ay opens upward instead. Do not take the coefficient 8 as a: it equals 4a.
  6. For the ellipse, the bigger denominator decides the major axis. In \frac{x^2}{25}+\frac{y^2}{9}=1 we have a=5, b=3, c^2=a^2-b^2=16, e=\frac45, foci (\pm4,0). The latus rectum is \frac{2b^2}{a}=3.6 and the sum of focal distances is 2a=10. Always e<1.
  7. For the hyperbola, c^2=a^2+b^2 and the sign between the terms is minus. In \frac{x^2}{16}-\frac{y^2}{9}=1 we get c=5, e=\frac54=1.25 and asymptotes y=\pm\frac34x. A rectangular hyperbola has a=b, so e=\sqrt2. Do not use the ellipse relation a^2-b^2 here.
  8. Name the conic from the general equation. Equal coefficients of x^2 and y^2 give a circle; same sign but unequal give an ellipse; opposite signs give a hyperbola; only one squared term gives a parabola (Fig 10.3). A quick check is the eccentricity: 0, below 1, equal to 1, above 1.

10.3.1 Worked checks

Q. Find the centre and radius of x^2+y^2-6x+4y-12=0, and the length of the tangent from (3,6).

Answer. 2g=-6, 2f=4, so the centre is (3,-2) and r=\sqrt{9+4+12}=5. Tangent length =\sqrt{S_1}=\sqrt{9+36-18+24-12}=\sqrt{39}\approx6.24.

Check the distance from (3,6) to the centre is 8 and \sqrt{64-25}=\sqrt{39}.

Q. For y^2=12x find the focus, directrix and length of the latus rectum.

Answer. 4a=12, so a=3. The focus is (3,0), the directrix is x=-3 and the latus rectum is 4a=12. The point (3,6) lies on the curve since 36=12\times3, and it is an end of the latus rectum.

Q. Find the eccentricity and foci of \frac{x^2}{25}+\frac{y^2}{9}=1.

Answer. a=5, b=3, c=\sqrt{25-9}=4, so e=\frac45=0.8 and the foci are (\pm4,0). Check: the sum of distances from (0,3) to the foci is 5+5=10=2a.

Formula and fact sheet

Item Result
Circle general centre (-g,-f), r=\sqrt{g^2+f^2-c}
Parabola y^2=4ax focus (a,0), directrix x=-a
Ellipse e^2=1-\frac{b^2}{a^2}
Hyperbola e^2=1+\frac{b^2}{a^2}

Chapter summary

  • Complete the square or use g,f,c to read a circle off its equation.
  • For y^2=4ax read a from the coefficient: y^2=12x gives a=3.
  • Ellipse: e<1; hyperbola: e>1.
  • Major axis follows the larger denominator.
  • Tangent length is \sqrt{S_1}, the power of the point.

Practice questions

Set A — Recall (20 questions)

  1. The radius of the circle x^2+y^2=25 isa25b10c5d2.5
  2. The centre of (x-2)^2+(y+3)^2=16 isa(-2,3)b(2,-3)c(2,3)d(-2,-3)
  3. The focus of y^2=4x isa(4,0)b(0,1)c(2,0)d(1,0)
  4. The eccentricity of a parabola isa1b0cless than 1dgreater than 1
  5. The directrix of y^2=8x isax=2bx=-2cx=-8dy=-2
  6. The length of the latus rectum of y^2=12x isa3b6c12d24
  7. The centre of x^2+y^2-4x+6y-12=0 isa(2,-3)b(-2,3)c(4,-6)d(-4,6)
  8. The eccentricity of a circle isa1bgreater than 1c0.5d0
  9. The eccentricity of a hyperbola isa0b1cgreater than 1dless than 1
  10. The length of the major axis of \frac{x^2}{25}+\frac{y^2}{9}=1 isa5b10c6d3
  11. The foci of \frac{x^2}{25}+\frac{y^2}{9}=1 area(\pm4,0)b(\pm5,0)c(\pm3,0)d(0,\pm4)
  12. The circle with centre the origin and radius 3 isax^2+y^2=3bx+y=9cx^2-y^2=9dx^2+y^2=9
  13. The vertex of y^2=4ax isa(a,0)b(-a,0)c(0,0)d(0,a)
  14. The asymptotes of \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 areay=\pm\frac{b}{a}xby=\pm\frac{a}{b}xcy=\pm xdx=\pm a
  15. The radius of x^2+y^2+2gx+2fy+c=0 isag^2+f^2-cb\sqrt{g^2+f^2-c}c\sqrt{g^2+f^2+c}dg+f-c
  16. The parabola x^2=4ay with a>0 opensato the rightbto the leftcdownwarddupward
  17. Which point lies on x^2+y^2=25?a(2,3)b(4,4)c(3,4)d(5,5)
  18. For \frac{x^2}{16}+\frac{y^2}{25}=1 the major axis lies along theay-axisbx-axiscline y=xdnone: it is a circle
  19. The focus of x^2=8y isa(2,0)b(0,2)c(0,8)d(8,0)
  20. The curve xy=4 representsaa parabolaban ellipseca circleda rectangular hyperbola
Answers and explanations — Set A

1. (c) — r² = 25.

2. (b) — centre is (h, k) with the signs reversed inside the brackets.

3. (d) — 4a = 4, a = 1.

4. (a) — by definition.

5. (b) — 4a = 8, a = 2, directrix x = −2.

6. (c) — the latus rectum is 4a = 12.

7. (a) — 2g = −4 and 2f = 6 give (−g, −f) = (2, −3).

8. (d) — a circle is an ellipse with a = b.

9. (c) — hyperbola has e > 1.

10. (b) — 2a = 10.

11. (a) — c² = 25 − 9 = 16.

12. (d) — x² + y² = r².

13. (c) — the vertex is the origin.

14. (a) — slope b/a.

15. (b) — remember the square root.

16. (d) — x² = 4ay opens up when a > 0.

17. (c) — 9 + 16 = 25.

18. (a) — 25 is the larger denominator, under y².

19. (b) — 4a = 8, a = 2, focus (0, 2).

20. (d) — xy = c² is a rectangular hyperbola.

Set B — Understanding (12 questions)

  1. The circle with centre (3,4) passing through the origin isax^2+y^2-6x-8y=0bx^2+y^2-6x-8y+25=0cx^2+y^2=25dx^2+y^2+6x+8y=0
  2. The radius of x^2+y^2-6x+8y-11=0 isa11b5c36d6
  3. The line y=5 and the circle x^2+y^2=25ado not meetbcut at two pointsctouch each otherdmeet at the centre
  4. The parabola with vertex at the origin and focus (3,0) isay^2=3xby^2=12xcx^2=12ydy^2=6x
  5. The eccentricity of \frac{x^2}{25}+\frac{y^2}{16}=1 isa\frac{4}{5}b\frac{5}{3}c\frac{9}{25}d\frac{3}{5}
  6. The eccentricity of \frac{x^2}{9}-\frac{y^2}{16}=1 isa\frac{5}{3}b\frac{3}{5}c\frac{4}{3}d\frac{25}{9}
  7. The latus rectum of \frac{x^2}{25}+\frac{y^2}{9}=1 has lengtha5b\frac{9}{5}c\frac{18}{5}d10
  8. The tangent to x^2+y^2=25 at (3,4) isa4x+3y=25b3x+4y=25c3x-4y=25dx+y=7
  9. The distance between the foci of \frac{x^2}{25}+\frac{y^2}{9}=1 isa4b5c10d8
  10. A circle has the join of (0,0) and (6,8) as a diameter. Its radius isa5b10c7d14
  11. The area of x^2+y^2=49 (take \pi=\frac{22}{7}) isa44b154c308d49
  12. On y^2=4x, the positive value of y when x=9 isa3b9c6d36
Answers and explanations — Set B

1. (a) — (x − 3)² + (y − 4)² = 25 expands to this with c = 0.

2. (d) — g = −3, f = 4: √(9 + 16 + 11) = 6.

3. (c) — the distance from the centre is 5 = r, so tangent.

4. (b) — a = 3 so y² = 12x.

5. (d) — √(1 − 16/25) = 3/5.

6. (a) — √(1 + 16/9) = 5/3.

7. (c) — 2b²/a = 18/5.

8. (b) — xx₁ + yy₁ = r².

9. (d) — 2ae = 2 × 4 = 8.

10. (a) — diameter = √(36 + 64) = 10.

11. (b) — (22/7) × 49 = 154.

12. (c) — y² = 36.

Set C — Application (8 questions)

  1. The radius of x^2+y^2+4x-2y-4=0 isa9b2c4d3
  2. The length of the tangent from (5,0) to x^2+y^2=9 isa4b5c16d3
  3. A circle of radius 3 lies in the first quadrant touching both axes. Its centre is at distance from the origina3b6c3\sqrt{2}d9
  4. On y^2=16x, the distance of the point with x=5 from the focus isa4b9c5d20
  5. The area of the ellipse \frac{x^2}{4}+y^2=1 isa2\pib4\pic\pid\frac{\pi}{2}
  6. The angle between the asymptotes of x^2-y^2=1 isa30°b45°c60°d90°
  7. The number of tangents that can be drawn to a circle from a point outside it isa1b2c3d0
  8. The point (2,2) with respect to the circle x^2+y^2=9 isaoutsidebon the circlecinsidedcannot be said
Answers and explanations — Set C

1. (d) — g = 2, f = −1: √(4 + 1 + 4) = 3.

2. (a) — √(25 − 9) = 4.

3. (c) — centre (3, 3), distance 3√2.

4. (b) — focal distance = x + a = 5 + 4.

5. (a) — π × 2 × 1.

6. (d) — asymptotes y = ±x are perpendicular.

7. (b) — two tangents from an external point.

8. (c) — 4 + 4 = 8 < 9.