Training Conic Sections Circles & Parabolas
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Circles & Parabolas

24 min Conic Sections

Circles & Parabolas

Circle

A circle with center $(h,k)$ and radius $r$:

$$(x-h)^2 + (y-k)^2 = r^2$$

Parabola

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).

Vertical axis: $(x-h)^2 = 4p(y-k)$, focus at $(h, k+p)$.

Horizontal axis: $(y-k)^2 = 4p(x-h)$, focus at $(h+p, k)$.

Completing the Square

To convert $x^2 + y^2 + Dx + Ey + F = 0$ to standard form, group and complete the square for each variable.

Example 1

Find the center and radius: $x^2 + y^2 - 6x + 4y - 12 = 0$.

$(x^2-6x+9)+(y^2+4y+4) = 12+9+4$

$(x-3)^2+(y+2)^2 = 25$. Center $(3,-2)$, $r = 5$.

Example 2

Find the vertex, focus, and directrix of $y = \frac{1}{8}x^2$.

$x^2 = 8y$, so $4p = 8$, $p = 2$. Vertex $(0,0)$, focus $(0,2)$, directrix $y = -2$.

Example 3

Write the equation of a circle with center $(-1,3)$ and radius $4$.

$$(x+1)^2 + (y-3)^2 = 16$$

Practice Problems

1. Find center and radius: $(x-2)^2+(y+1)^2 = 9$.
2. Write the equation of a circle: center $(0,0)$, $r = 7$.
3. Complete the square: $x^2+y^2+2x-8y+8=0$.
4. Find vertex and focus: $x^2 = 12y$.
5. Find vertex and focus: $(y-1)^2 = -8(x+2)$.
6. Write the parabola with vertex $(0,0)$ and focus $(0,3)$.
7. What is the directrix of $x^2 = -4y$?
8. A satellite dish is parabolic: 10 ft wide, 2 ft deep. How far is the focus from the vertex?
9. Find the radius: $x^2+y^2 = 50$.
10. Does $(x+3)^2 = 16(y-1)$ open up or down?
11. Find the equation of a circle passing through $(3,4)$ with center at the origin.
12. Identify: $x^2 + y^2 - 10x + 6y + 18 = 0$ — circle or parabola?
Show Answer Key

1. Center $(2,-1)$, $r = 3$

2. $x^2+y^2=49$

3. $(x+1)^2+(y-4)^2 = 9$. Center $(-1,4)$, $r=3$.

4. Vertex $(0,0)$, $4p=12$, $p=3$; focus $(0,3)$

5. Vertex $(-2,1)$, $p=-2$; focus $(-4,1)$; opens left

6. $x^2 = 12y$

7. $y = 1$ (since $p = -1$, directrix is $y = -p = 1$)

8. $x^2 = 4py$; at edge: $5^2 = 4p(2)$; $p = 25/8 = 3.125$ ft

9. $r = \sqrt{50} = 5\sqrt{2}$

10. $4p = 16 > 0$, opens up

11. $r = \sqrt{9+16} = 5$; $x^2+y^2=25$

12. Circle (both $x^2$ and $y^2$ with same coefficient)

Ellipses