Conic Sections — Challenge
Slice a cone four ways and you get every curve in this unit.
What these problems 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 does x2+y2=9 describe?
- A circle
- An ellipse
- A parabola
- A hyperbola
- A straight line
Worked answer
A circle
Both variables are squared, both have the same coefficient, and they are added.
That is the equation of a circle centred at the origin, with r2 = 9 so r = 3.
Equal coefficients are what make it a circle rather than an ellipse.
Problem 2
What curve does y=x2 describe?
- A circle
- An ellipse
- A parabola
- A hyperbola
- A straight line
Worked answer
A parabola
Only one of the two variables is squared here.
A conic with exactly one squared variable is a parabola.
This one opens upwards with its vertex at the origin.
Problem 3
What curve does x29+y24=1 describe?
- A circle
- An ellipse
- A parabola
- A hyperbola
- A straight line
Worked answer
An ellipse
Both variables are squared and added, which points to a circle or an ellipse.
The two denominators differ, so the curve is wider than it is tall: it stretches to 3 along x and 2 along y.
Unequal denominators make it an ellipse; equal ones would give a circle.
Problem 4
What curve does x29-y24=1 describe?
- A circle
- An ellipse
- A parabola
- A hyperbola
- A straight line
Worked answer
A hyperbola
Both variables are squared, but here they are subtracted rather than added.
A difference of squared terms gives a hyperbola, which comes in two separate branches.
The minus sign is the whole difference between this and the ellipse.
Problem 5
What is the radius of the circle x2+y2=49?
- 7
- 49
- 14
- 24.5
- 3.5
Worked answer
7
For a circle at the origin the equation is x2+y2 = r2.
So r2 = 49 and r = 7.
The diameter would be 14, and taking 49 itself is the usual error.
Problem 6
The ellipse x225 + y29 = 1 fits snugly inside a rectangle. What is that rectangle’s perimeter?
- 32
- 30
- 16
- 34
- 64
Worked answer
32
The curve reaches √25 = 5 either side horizontally, so the rectangle is 10 wide. It reaches √9 = 3 either side vertically, so the rectangle is 6 tall. Perimeter = 2(10+6) = 32.
Problem 7
How many points do x2+y2 = 25 and y = 5 share?
- 1
- 0
- 2
- 4
- Infinitely many
Worked answer
1
Substitute y = 5: x2 + 25 = 25, so x2 = 0. That has the single solution x = 0. The line touches the circle at exactly one point, (0,5) — it is the tangent at the top.
Problem 8
A parabola has vertex at the origin and passes through (2, 8). If its equation is y = ax2, what is a?
- 2
- 4
- 8
- 12
- 16
Worked answer
2
Substitute the point: 8 = a(2)2. 8 = 4a. So a = 2, and the parabola is y = 2x2 — twice as steep as y = x2.