Conic sections

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1. Circles: equation, center and radius

Warm-up

Three quick questions from earlier lessons.

Discover

A point (x,y)(x, y) is on the circle with center (2,1)(2, 1) and radius 3 when its distance to (2,1)(2, 1) is 3. Which equation says that?

  1. (x−2)2+(y−1)2=9(x - 2)^2 + (y - 1)^2 = 9
  2. (x+2)2+(y+1)2=9(x + 2)^2 + (y + 1)^2 = 9
  3. (x−2)2+(y−1)2=3(x - 2)^2 + (y - 1)^2 = 3

Key idea

A circle is the set of points at a fixed distance rr from a center (h,k)(h, k).

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2. From the expanded form x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0, complete the square in xx and in yy to read the center and radius.

Worked example

Find the center and radius of x2+y2−4x+6y−3=0x^2 + y^2 - 4x + 6y - 3 = 0.

(x−2)2−4+(y+3)2−9−3=0(x - 2)^2 - 4 + (y + 3)^2 - 9 - 3 = 0
Group and complete the squares: x² − 4x = (x − 2)² − 4, y² + 6y = (y + 3)² − 9.
(x−2)2+(y+3)2=16(x - 2)^2 + (y + 3)^2 = 16
Move the numbers to the right.
Center (2,−3)(2, -3), radius 44
Read h, k and r = √16.

Your turn, guided

Your turn: (x+5)2+(y−1)2=49(x + 5)^2 + (y - 1)^2 = 49.

Center (−5,1)(-5, 1)
x + 5 = x − (−5).
Radius 77
√49.

Practice ★

  1. What is the radius of the circle (x−9)2+(y+1)2=16(x - 9)^2 + (y + 1)^2 = 16?
  2. What is the radius of the circle (x−1)2+(y+7)2=64(x - 1)^2 + (y + 7)^2 = 64?
  3. What is the radius of the circle (x−4)2+(y+9)2=16(x - 4)^2 + (y + 9)^2 = 16?

Watch out!

Practice ★★

  1. Find the center and radius of the circle x2+y2−10x+4y+25=0x^2 + y^2 - 10x + 4y + 25 = 0.
  2. Find the center and radius of the circle x2+y2−2x+4y+−11=0x^2 + y^2 - 2x + 4y + -11 = 0.
  1. Which is the equation of the circle with center (2,−8)(2, -8) and radius 7?
    1. (x−2)2+(y+8)2=49(x - 2)^2 + (y + 8)^2 = 49
    2. (x+2)2+(y−8)2=49(x + 2)^2 + (y - 8)^2 = 49
    3. (x−2)2+(y+8)2=7(x - 2)^2 + (y + 8)^2 = 7
  2. Which is the equation of the circle with center (7,−2)(7, -2) and radius 9?
    1. (x−7)2+(y+2)2=81(x - 7)^2 + (y + 2)^2 = 81
    2. (x+7)2+(y−2)2=81(x + 7)^2 + (y - 2)^2 = 81
    3. (x−7)2+(y+2)2=9(x - 7)^2 + (y + 2)^2 = 9

Exit ticket

Three questions to check you have got it.

  1. Which is the equation of the circle with center (2,−8)(2, -8) and radius 7?
    1. (x−2)2+(y+8)2=49(x - 2)^2 + (y + 8)^2 = 49
    2. (x+2)2+(y−8)2=49(x + 2)^2 + (y - 8)^2 = 49
    3. (x−2)2+(y+8)2=7(x - 2)^2 + (y + 8)^2 = 7
  2. Which is the equation of the circle with center (7,−2)(7, -2) and radius 9?
    1. (x−7)2+(y+2)2=81(x - 7)^2 + (y + 2)^2 = 81
    2. (x+7)2+(y−2)2=81(x + 7)^2 + (y - 2)^2 = 81
    3. (x−7)2+(y+2)2=9(x - 7)^2 + (y + 2)^2 = 9
  3. Which is the equation of the circle with center (6,−1)(6, -1) and radius 6?
    1. (x−6)2+(y+1)2=36(x - 6)^2 + (y + 1)^2 = 36
    2. (x+6)2+(y−1)2=36(x + 6)^2 + (y - 1)^2 = 36
    3. (x−6)2+(y+1)2=6(x - 6)^2 + (y + 1)^2 = 6

Summary card

Circles

  • (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2: center (h,k)(h, k), radius rr.
  • Expanded form: complete the square in xx and in yy.

Watch out: (x+3)(x + 3) means h=−3h = -3, and the right side is r2r^2.

2. Parabolas: focus and directrix

Warm-up

Three quick questions from earlier lessons.

Discover

A satellite dish is a parabola: every signal coming straight in bounces to one point, the focus. For x2=8yx^2 = 8y, where is it?

  1. (0,2)(0, 2)
  2. (0,8)(0, 8)

Key idea

A parabola is the set of points as far from a point (the focus) as from a line (the directrix).

x2=4pyx^2 = 4py: vertex (0,0)(0, 0), focus (0,p)(0, p), directrix y=−py = -p, opens up if p>0p > 0. y2=4pxy^2 = 4px: focus (p,0)(p, 0), directrix x=−px = -p, opens right if p>0p > 0.

Worked example

Find the focus and directrix of y2=−12xy^2 = -12x.

4p=−124p = -12, so p=−3p = -3
Match with y² = 4px.
Focus (−3,0)(-3, 0): it opens to the left
Focus (p, 0).
Directrix x=3x = 3
Directrix x = −p.

Your turn, guided

Your turn: x2=20yx^2 = 20y.

p=5p = 5, focus (0,5)(0, 5)
4p = 20.
Directrix y=−5y = -5
y = −p: type the number.

Practice ★

  1. The parabola x2=8yx^2 = 8y has its focus at (0,c)(0, c). What is cc?
  2. The parabola x2=24yx^2 = 24y has its focus at (0,c)(0, c). What is cc?
  3. The parabola x2=4yx^2 = 4y has its focus at (0,c)(0, c). What is cc?

Watch out!

Practice ★★

  1. The parabola y2=8xy^2 = 8x has the directrix x=dx = d. What is dd?
  2. The parabola y2=20xy^2 = 20x has the directrix x=dx = d. What is dd?
  3. The parabola y2=4xy^2 = 4x has the directrix x=dx = d. What is dd?
  1. A parabola has focus (0,3)(0, 3) and directrix y=−3y = -3. Its equation is x2=kyx^2 = ky. What is kk?
  2. A parabola has focus (0,2)(0, 2) and directrix y=−2y = -2. Its equation is x2=kyx^2 = ky. What is kk?

Exit ticket

Three questions to check you have got it.

  1. The parabola y2=8xy^2 = 8x has the directrix x=dx = d. What is dd?
  2. The parabola y2=20xy^2 = 20x has the directrix x=dx = d. What is dd?
  3. The parabola y2=4xy^2 = 4x has the directrix x=dx = d. What is dd?

Summary card

Parabolas

  • x2=4pyx^2 = 4py: focus (0,p)(0, p), directrix y=−py = -p.
  • y2=4pxy^2 = 4px: focus (p,0)(p, 0), directrix x=−px = -p.

Watch out: divide by 4 to get pp.

3. Ellipses: axes, vertices and foci

Warm-up

Three quick questions from earlier lessons.

Discover

x225+y29=1\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1. Where does this ellipse cross the xx-axis?

  1. At x=±5x = \pm 5
  2. At x=±25x = \pm 25

Key idea

An ellipse is the set of points whose distances to two foci add up to a constant.

x2a2+y2b2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 with a>ba > b: vertices (±a,0)(\pm a, 0), co-vertices (0,±b)(0, \pm b), foci (±c,0)(\pm c, 0) with c2=a2−b2c^2 = a^2 - b^2. The larger denominator shows the major axis: under y2y^2, the ellipse is vertical.

Worked example

Describe x216+y225=1\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1.

Vertical major axis, a=5a = 5, b=4b = 4
25 > 16, under y².
Vertices (0,±5)(0, \pm 5)
Vertices on the y-axis.
Foci (0,±3)(0, \pm 3)
c² = 25 − 16.

Your turn, guided

Your turn: x2100+y236=1\dfrac{x^2}{100} + \dfrac{y^2}{36} = 1.

Vertices (±10,0)(\pm 10, 0)
100 > 36, under x²: a = 10, b = 6.
Foci (±8,0)(\pm 8, 0)
c² = 100 − 36 = 64.

Practice ★

  1. The ellipse x236+y21=1\dfrac{x^2}{ 36 } + \dfrac{y^2}{ 1 } = 1 has its vertices at (±c,0)(\pm c, 0). What is cc?
  2. The ellipse x225+y21=1\dfrac{x^2}{ 25 } + \dfrac{y^2}{ 1 } = 1 has its vertices at (±c,0)(\pm c, 0). What is cc?
  3. The ellipse x29+y24=1\dfrac{x^2}{ 9 } + \dfrac{y^2}{ 4 } = 1 has its vertices at (±c,0)(\pm c, 0). What is cc?

Watch out!

Practice ★★

  1. The ellipse x225+y216=1\dfrac{x^2}{ 25 } + \dfrac{y^2}{ 16 } = 1 has foci at (±c,0)(\pm c, 0). Find cc.
  2. The ellipse x2169+y2144=1\dfrac{x^2}{ 169 } + \dfrac{y^2}{ 144 } = 1 has foci at (±c,0)(\pm c, 0). Find cc.
  3. The ellipse x2289+y264=1\dfrac{x^2}{ 289 } + \dfrac{y^2}{ 64 } = 1 has foci at (±c,0)(\pm c, 0). Find cc.
  1. Is the major axis of x21+y236=1\dfrac{x^2}{ 1 } + \dfrac{y^2}{ 36 } = 1 horizontal or vertical?
    1. Horizontal
    2. Vertical
    3. Neither: it is a circle
  2. Is the major axis of x24+y264=1\dfrac{x^2}{ 4 } + \dfrac{y^2}{ 64 } = 1 horizontal or vertical?
    1. Horizontal
    2. Vertical
    3. Neither: it is a circle

Exit ticket

Three questions to check you have got it.

  1. The ellipse x225+y216=1\dfrac{x^2}{ 25 } + \dfrac{y^2}{ 16 } = 1 has foci at (±c,0)(\pm c, 0). Find cc.
  2. The ellipse x2169+y2144=1\dfrac{x^2}{ 169 } + \dfrac{y^2}{ 144 } = 1 has foci at (±c,0)(\pm c, 0). Find cc.
  3. The ellipse x2289+y264=1\dfrac{x^2}{ 289 } + \dfrac{y^2}{ 64 } = 1 has foci at (±c,0)(\pm c, 0). Find cc.

Summary card

Ellipses

  • x2a2+y2b2=1\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1: vertices at distance aa, co-vertices at bb.
  • Foci: c2=a2−b2c^2 = a^2 - b^2 (larger minus smaller).
  • The larger denominator gives the major axis.

Watch out: minus for ellipses, plus for hyperbolas.

4. Hyperbolas: vertices, foci and asymptotes

Warm-up

Three quick questions from earlier lessons.

Discover

x29−y216=1\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1. Does this curve cross the yy-axis?

  1. No
  2. Yes, at y=±4y = \pm 4

Key idea

A hyperbola is the set of points whose distances to two foci differ by a constant.

x2a2−y2b2=1\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1: vertices (±a,0)(\pm a, 0), foci (±c,0)(\pm c, 0) with c2=a2+b2c^2 = a^2 + b^2, asymptotes y=±baxy = \pm\dfrac{b}{a}x. If the y2y^2 term is the positive one, the branches open up and down: vertices (0,±a)(0, \pm a).

Worked example

Describe x29−y216=1\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1.

Vertices (±3,0)(\pm 3, 0)
x² is positive: a² = 9, b² = 16.
Foci (±5,0)(\pm 5, 0)
c² = 9 + 16.
Asymptotes y=±43xy = \pm\dfrac{4}{3}x
Slope b/a = 4/3.

Your turn, guided

Your turn: y225−x2144=1\dfrac{y^2}{25} - \dfrac{x^2}{144} = 1.

Vertices (0,±5)(0, \pm 5)
y² is positive: vertices on the y-axis, a = 5.
Foci (0,±13)(0, \pm 13)
c² = 25 + 144 = 169.

Practice ★

  1. The hyperbola y264−x225=1\dfrac{y^2}{ 64 } - \dfrac{x^2}{ 25 } = 1 has its vertices at (0,±c)(0, \pm c). What is cc?
  2. The hyperbola y216−x225=1\dfrac{y^2}{ 16 } - \dfrac{x^2}{ 25 } = 1 has its vertices at (0,±c)(0, \pm c). What is cc?
  3. The hyperbola y225−x236=1\dfrac{y^2}{ 25 } - \dfrac{x^2}{ 36 } = 1 has its vertices at (0,±c)(0, \pm c). What is cc?

Watch out!

Practice ★★

  1. The hyperbola x2144−y2256=1\dfrac{x^2}{ 144 } - \dfrac{y^2}{ 256 } = 1 has foci at (±c,0)(\pm c, 0). Find cc.
  2. The hyperbola x2225−y264=1\dfrac{x^2}{ 225 } - \dfrac{y^2}{ 64 } = 1 has foci at (±c,0)(\pm c, 0). Find cc.
  1. The hyperbola x21−y236=1\dfrac{x^2}{ 1 } - \dfrac{y^2}{ 36 } = 1 has asymptotes y=±mxy = \pm mx. What is mm?
  2. The hyperbola x216−y264=1\dfrac{x^2}{ 16 } - \dfrac{y^2}{ 64 } = 1 has asymptotes y=±mxy = \pm mx. What is mm?
  3. The hyperbola x281−y236=1\dfrac{x^2}{ 81 } - \dfrac{y^2}{ 36 } = 1 has asymptotes y=±mxy = \pm mx. What is mm?

Exit ticket

Three questions to check you have got it.

  1. The hyperbola x21−y236=1\dfrac{x^2}{ 1 } - \dfrac{y^2}{ 36 } = 1 has asymptotes y=±mxy = \pm mx. What is mm?
  2. The hyperbola x216−y264=1\dfrac{x^2}{ 16 } - \dfrac{y^2}{ 64 } = 1 has asymptotes y=±mxy = \pm mx. What is mm?
  3. The hyperbola x281−y236=1\dfrac{x^2}{ 81 } - \dfrac{y^2}{ 36 } = 1 has asymptotes y=±mxy = \pm mx. What is mm?

Summary card

Hyperbolas

  • Vertices on the axis of the positive term, at distance aa.
  • Foci: c2=a2+b2c^2 = a^2 + b^2.
  • x2a2−y2b2=1\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1: asymptotes y=±baxy = \pm\dfrac{b}{a}x.

Watch out: plus for hyperbolas, minus for ellipses.

5. Identify a conic from its equation

Warm-up

Three quick questions from earlier lessons.

Discover

4x2+4y2=364x^2 + 4y^2 = 36. Which conic is it?

  1. A circle
  2. A hyperbola

Key idea

The squared terms tell which conic an equation draws.

Only one variable squared: parabola. Both squared, same sign: ellipse (a circle if the coefficients are equal). Both squared, opposite signs: hyperbola.

Worked example

Identify 9y2−x2=99y^2 - x^2 = 9, then put it in standard form.

A hyperbola
Both squared, opposite signs.
y2−x29=1y^2 - \dfrac{x^2}{9} = 1
Divide by 9.
Branches open up and down, vertices (0,±1)(0, \pm 1)
y² is positive.

Your turn, guided

Your turn: x2+4y2=16x^2 + 4y^2 = 16.

An ellipse
Both squared, same sign, different coefficients.
a=4a = 4: vertices (±4,0)(\pm 4, 0)
Divide by 16: x²/16 + y²/4 = 1. Type a.

Practice ★

  1. What kind of conic is 2x2+3y2=362x^2 + 3y^2 = 36?
    1. An ellipse
    2. A circle
    3. A hyperbola
    4. A parabola
  2. What kind of conic is 2x2+7y2=672x^2 + 7y^2 = 67?
    1. An ellipse
    2. A circle
    3. A hyperbola
    4. A parabola

Watch out!

Practice ★★

  1. What kind of conic is 9x2−5y2=139x^2 - 5y^2 = 13?
    1. A hyperbola
    2. An ellipse
    3. A parabola
  2. What kind of conic is 4x2−2y2=404x^2 - 2y^2 = 40?
    1. A hyperbola
    2. An ellipse
    3. A parabola
  1. What kind of conic is y=5x2−9xy = 5x^2 - 9x?
    1. A parabola
    2. A hyperbola
    3. A circle
  2. What kind of conic is y=9x2−4xy = 9x^2 - 4x?
    1. A parabola
    2. A hyperbola
    3. A circle

Exit ticket

Three questions to check you have got it.

  1. What kind of conic is 2x2+3y2=362x^2 + 3y^2 = 36?
    1. An ellipse
    2. A circle
    3. A hyperbola
    4. A parabola
  2. What kind of conic is 2x2+7y2=672x^2 + 7y^2 = 67?
    1. An ellipse
    2. A circle
    3. A hyperbola
    4. A parabola
  3. What kind of conic is 6x2+5y2=786x^2 + 5y^2 = 78?
    1. An ellipse
    2. A circle
    3. A hyperbola
    4. A parabola

Summary card

Which conic?

  • One variable squared: parabola.
  • Both, same sign: ellipse (equal coefficients: circle).
  • Both, opposite signs: hyperbola.

Watch out: look at the squared terms, not the constant.

6. Systems with a line and a conic

Warm-up

Three quick questions from earlier lessons.

Discover

How many points can a line and a circle have in common?

  1. 0, 1 or 2
  2. Always 2

Key idea

To find where a line meets a conic, solve the two equations together.

Substitute the line into the conic: you get a quadratic in one variable. Its real solutions give the intersection points: 2 solutions, 2 points; a double root, a tangent; no real root, no intersection. Then find the other coordinate from the line.

Worked example

Where does y=x+1y = x + 1 meet x2+y2=25x^2 + y^2 = 25?

x2+(x+1)2=25x^2 + (x + 1)^2 = 25
Substitute y.
2x2+2x−24=02x^2 + 2x - 24 = 0, so x2+x−12=0x^2 + x - 12 = 0
Expand and simplify.
(3,4)(3, 4) and (−4,−3)(-4, -3)
(x + 4)(x − 3) = 0, then y = x + 1.

Your turn, guided

Your turn: where does y=3y = 3 meet x2+y2=25x^2 + y^2 = 25?

x2=16x^2 = 16
Substitute y = 3.
x=4x = 4 or x=−4x = -4
Two square roots: type the negative one.

Practice ★

  1. The circle x2+y2=169x^2 + y^2 = 169 and the line y=12y = 12 meet at two points. Give their xx-coordinates.
  2. The circle x2+y2=25x^2 + y^2 = 25 and the line y=4y = 4 meet at two points. Give their xx-coordinates.
  3. The circle x2+y2=400x^2 + y^2 = 400 and the line y=16y = 16 meet at two points. Give their xx-coordinates.

Watch out!

Practice ★★

  1. How many points do the circle x2+y2=16x^2 + y^2 = 16 and the line y=11y = 11 have in common?
    1. None
    2. One
    3. Two
  1. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+6y = x + 6.
  2. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+20y = x + 20.
  3. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+2y = x + 2.

Exit ticket

Three questions to check you have got it.

  1. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+6y = x + 6.
  2. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+20y = x + 20.
  3. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+2y = x + 2.

Summary card

A line and a conic

  1. Substitute the line into the conic.
  2. Solve the quadratic: 0, 1 or 2 real solutions.
  3. Find the other coordinate from the line.

Watch out: a negative square means no intersection.

7. Tangent lines to a circle

Warm-up

Three quick questions from earlier lessons.

Discover

A wheel rolls on the ground. The ground touches it at one point. What angle does the spoke to that point make with the ground?

  1. 90∘90^\circ
  2. It depends on the speed

Key idea

Tangents and radii are perpendicular.

A tangent touches the circle at one point and is perpendicular to the radius there. The two tangent segments from an outside point are equal in length.

Worked example

Radius 6, point PP at distance 10 from the center.

Right triangle with hypotenuse 10, leg 6
Right angle at the point of contact.
Tangent length 8
√(100 − 36).

Your turn, guided

Your turn: radius 5, distance 13. The tangent length.

1212
√(169 − 25).

Practice ★

  1. A circle has radius 6. Point PP is 10 from the center. Find the length of a tangent segment from PP to the circle.
  2. A circle has radius 3. Point PP is 5 from the center. Find the length of a tangent segment from PP to the circle.
  1. A line touches a circle of radius 7 at exactly one point TT. What is the angle between the line and the radius to TT?
    1. 90∘90^\circ
    2. It depends on the radius
    3. 0∘0^\circ: they lie along each other

Watch out!

Practice ★★

  1. Two tangents from PP touch a circle (center OO) at AA and BB, with ∠APB=61∘\angle APB = 61^\circ. Find ∠AOB\angle AOB (degrees).
  2. Two tangents from PP touch a circle (center OO) at AA and BB, with ∠APB=76∘\angle APB = 76^\circ. Find ∠AOB\angle AOB (degrees).

Exit ticket

Three questions to check you have got it.

  1. A circle has radius 6. Point PP is 10 from the center. Find the length of a tangent segment from PP to the circle.
  2. A circle has radius 3. Point PP is 5 from the center. Find the length of a tangent segment from PP to the circle.
  3. A circle has radius 12. Point PP is 20 from the center. Find the length of a tangent segment from PP to the circle.

Summary card

Tangent lines

Tangent ⊥ radius; tangent segments from a point are equal.

Watch out: a right triangle, not a subtraction.

8. Volume of spheres; density

Warm-up

Three quick questions from earlier lessons.

Discover

A ball of radius rr fits exactly in a cylinder of radius rr and height 2r2r. The ball fills 23\tfrac{2}{3} of the cylinder. What is the ball's volume?

Not to scale
  1. 43πr3\tfrac{4}{3}\pi r^3
  2. 4πr24\pi r^2

Key idea

Spheres, and how heavy a solid is.

Sphere: V=43πr3V = \tfrac{4}{3}\pi r^3 (surface area 4πr24\pi r^2). Density = mass ÷ volume, so mass = density × volume.

Worked example

A steel ball of radius 3 cm, density 7.8 g/cm³.

V=36π≈113.1V = 36\pi \approx 113.1 cm³
(4/3)π · 27.
m≈7.8×113.1≈882m \approx 7.8 \times 113.1 \approx 882 g
Mass = density × volume.

Your turn, guided

Your turn: mass 200 g, volume 50 cm³. The density.

44 g/cm³
200 ÷ 50.

Practice ★

  1. Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
  2. Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale
  1. A block has mass 526 g and volume 27 cm³. Find its density (g/cm³).

Watch out!

Practice ★★

  1. Find the volume of a hemisphere (half a ball) of radius 3 cm (exact, in terms of π).
  2. Find the volume of a hemisphere (half a ball) of radius 5 cm (exact, in terms of π).

Exit ticket

Three questions to check you have got it.

  1. Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
  2. Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale
  3. Find the volume of a sphere of radius 8 cm (exact, in terms of π).Not to scale

Summary card

Spheres and density

V=43πr3V = \tfrac{4}{3}\pi r^3; density = mass/volume.

Watch out: r3r^3 for volume, r2r^2 for area.

Answers: Conic sections

1. Circles: equation, center and radius

A

  1. 44
  2. 88
  3. 44

B

  1. 22
  2. 44

C

  1. (x−2)2+(y+8)2=49(x - 2)^2 + (y + 8)^2 = 49
  2. (x−7)2+(y+2)2=81(x - 7)^2 + (y + 2)^2 = 81

D

  1. (x−2)2+(y+8)2=49(x - 2)^2 + (y + 8)^2 = 49
  2. (x−7)2+(y+2)2=81(x - 7)^2 + (y + 2)^2 = 81
  3. (x−6)2+(y+1)2=36(x - 6)^2 + (y + 1)^2 = 36

2. Parabolas: focus and directrix

A

  1. 22
  2. 66
  3. 11

B

  1. −2-2
  2. −5-5
  3. −1-1

C

  1. 1212
  2. 88

D

  1. −2-2
  2. −5-5
  3. −1-1

3. Ellipses: axes, vertices and foci

A

  1. 66
  2. 55
  3. 33

B

  1. 33
  2. 55
  3. 1515

C

  1. Vertical
  2. Vertical

D

  1. 33
  2. 55
  3. 1515

4. Hyperbolas: vertices, foci and asymptotes

A

  1. 88
  2. 44
  3. 55

B

  1. 2020
  2. 1717

C

  1. 66
  2. 22
  3. 23\frac{2}{3}

D

  1. 66
  2. 22
  3. 23\frac{2}{3}

5. Identify a conic from its equation

A

  1. An ellipse
  2. An ellipse

B

  1. A hyperbola
  2. A hyperbola

C

  1. A parabola
  2. A parabola

D

  1. An ellipse
  2. An ellipse
  3. An ellipse

6. Systems with a line and a conic

A

  1. 5, −55,\ -5
  2. 3, −33,\ -3
  3. 12, −1212,\ -12

B

  1. None

C

  1. 3, −23,\ -2
  2. 5, −45,\ -4
  3. 2, −12,\ -1

D

  1. 3, −23,\ -2
  2. 5, −45,\ -4
  3. 2, −12,\ -1

7. Tangent lines to a circle

A

  1. 88
  2. 44

B

  1. 90∘90^\circ

C

  1. 119119
  2. 104104

D

  1. 88
  2. 44
  3. 1616

8. Volume of spheres; density

A

  1. 500π3\frac{500 \pi}{3}
  2. 972π972 \pi

B

  1. 52627\frac{526}{27}

C

  1. 18π18 \pi
  2. 250π3\frac{250 \pi}{3}

D

  1. 500π3\frac{500 \pi}{3}
  2. 972π972 \pi
  3. 2048π3\frac{2048 \pi}{3}