Conic Sections — Practice
Slice a cone four ways and you get every curve in this unit.
What these questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Radius — Centre to edge. Half the diameter.
- Parabola — The U-shaped graph of a quadratic function.
- Vertex — A turning point of the curve, on its axis.
Problem 1
What curve is x2+y2=25?
- A circle
- An ellipse
- A parabola
- A hyperbola
Worked answer
A circle
Radius 5, centred at the origin
Problem 2
What curve is y = x2 - 4?
- A parabola
- A circle
- An ellipse
- A hyperbola
Worked answer
A parabola
Only x is squared, so it is a parabola
Problem 3
What curve is x29 - y24 = 1?
- A hyperbola
- An ellipse
- A circle
- A parabola
Worked answer
A hyperbola
Opposite signs open the curve out
Problem 4
Give the radius of x2+y2=49.
Worked answer
7
√49 = 7
Problem 5
Give the centre of (x-2)2+(y+5)2=9 as an ordered pair.
Worked answer
(2,-5)
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Problem 6
For x225+y216=1, how far does the curve reach along the x-axis?
Worked answer
5
√25 = 5
Problem 7
For the same ellipse, how far does it reach along the y-axis?
Worked answer
4
√16 = 4
Problem 8
Which way does y216-x29=1 open?
- Up and down
- Left and right
- All four ways
- It does not open
Worked answer
Up and down
The positive term names the opening direction
Problem 9
Give the vertex of y=(x-3)2+5 as an ordered pair.
Worked answer
(3,5)
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Problem 10
Find the radius of x2+y2-6x+4y-3=0.
Worked answer
4
(x-3)2+(y+2)2 = 3+9+4 = 16
r = 4
Problem 11
Find the centre of x2+y2-6x+4y-3=0 as an ordered pair.
Worked answer
(3,-2)
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Problem 12
What curve is 4x2+9y2=36?
- An ellipse
- A circle
- A hyperbola
- A parabola
Worked answer
An ellipse
Dividing by 36 gives x29+y24=1
Problem 13
For 4x2+9y2=36, how far does the curve reach along the x-axis?
Worked answer
3
x29+y24=1
√9 = 3
Problem 14
Which way does x = 2y2 open?
- Right
- Up
- Left
- Down
Worked answer
Right
y is squared, so the axis is horizontal
The positive coefficient opens it right
Problem 15
A circle has centre (1,-4) and radius 6. Give the constant on the right of its standard equation.
Worked answer
36
62 = 36
Problem 16
Changing the minus to a plus in x29-y24=1 turns it into what?
- An ellipse
- A parabola
- A circle
- A line
Worked answer
An ellipse
Two positive squared terms with different denominators give an ellipse