Circles and Parabolas

Standard forms and key features of circles and parabolas

Circles and Parabolas

Introduction to Conic Sections

Conic sections are curves formed by the intersection of a plane and a double cone. The four types are:

Circle
Parabola
Ellipse
Hyperbola

Circles

A circle is the set of all points equidistant from a center point.

Standard Form

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

Where:

$(h, k)$ = center
$r$ = radius

General Form

$x^2 + y^2 + Dx + Ey + F = 0$

Converting General to Standard: Complete the square for both $x$ and $y$ terms.

Key Features

Center: $(h, k)$
Radius: $r$ (distance from center to any point on circle)
Diameter: $2r$
Equation of circle with center at origin: $x^2 + y^2 = r^2$

Example

Circle with center $(3, -2)$ and radius $5$ : $(x - 3)^2 + (y + 2)^2 = 25$

Parabolas

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

Vertical Parabolas (opens up or down)

Standard Form (vertex form): $(x - h)^2 = 4p(y - k)$

Where:

$(h, k)$ = vertex
$p$ = distance from vertex to focus (and vertex to directrix)
If $p > 0$ : opens upward
If $p < 0$ : opens downward

Key features:

Vertex: $(h, k)$
Focus: $(h, k + p)$
Directrix: $y = k - p$
Axis of symmetry: $x = h$

Alternate form: $y = a(x - h)^2 + k$

Where $a = \frac{1}{4p}$

Horizontal Parabolas (opens left or right)

Standard Form: $(y - k)^2 = 4p(x - h)$

Where:

$(h, k)$ = vertex
If $p > 0$ : opens right
If $p < 0$ : opens left

Key features:

Vertex: $(h, k)$
Focus: $(h + p, k)$
Directrix: $x = h - p$
Axis of symmetry: $y = k$

Converting from General Form

$Ax^2 + Bx + Cy + D = 0$ (vertical parabola)

Complete the square on $x$ to get vertex form.

Example: Vertical Parabola

Vertex at $(2, 3)$ , opens upward, focus $2$ units above vertex: $(x - 2)^2 = 4(2)(y - 3)$ $(x - 2)^2 = 8(y - 3)$

Focus: $(2, 5)$ , Directrix: $y = 1$

Graphing Tips

For Circles:

Plot center $(h, k)$
Count $r$ units in all directions
Sketch smooth curve

For Parabolas:

Plot vertex $(h, k)$
Determine direction (up/down/left/right)
Plot focus and draw directrix
Sketch symmetric curve

📚 Practice Problems

1Problem 1easy

❓ Question:

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

💡 Show Solution

Convert to standard form by completing the square:

Step 1: Group $x$ and $y$ terms: $(x^2 - 6x) + (y^2 + 4y) = 12$

Step 2: Complete the square for $x$ :

Coefficient of $x$ : $-6$
Half of it: $-3$
Square it: $9$

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

Step 3: Complete the square for $y$ :

Coefficient of $y$ : $4$
Half of it: $2$
Square it: $4$

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

Step 4: Factor and simplify: $(x - 3)^2 + (y + 2)^2 = 25$

Step 5: Identify center and radius:

Standard form: $(x - h)^2 + (y - k)^2 = r^2$
Center: $(h, k) = (3, -2)$
Radius: $r = \sqrt{25} = 5$

Answers:

Center: $(3, -2)$
Radius: $5$

2Problem 2medium

❓ Question:

Write the equation of a parabola with vertex at $(-1, 4)$ and focus at $(-1, 6)$ .

💡 Show Solution

Determine parabola characteristics:

Step 1: Analyze vertex and focus:

Vertex: $(-1, 4)$
Focus: $(-1, 6)$
Same $x$ -coordinate → vertical parabola
Focus is above vertex → opens upward

Step 2: Find $p$ : $p = \text{distance from vertex to focus}$ $p = 6 - 4 = 2$

Step 3: Use standard form for vertical parabola: $(x - h)^2 = 4p(y - k)$

With $h = -1$ , $k = 4$ , and $p = 2$ : $(x - (-1))^2 = 4(2)(y - 4)$ $(x + 1)^2 = 8(y - 4)$

Step 4: Verify key features:

Vertex: $(-1, 4)$ ✓
Focus: $(-1, 4 + 2) = (-1, 6)$ ✓
Directrix: $y = 4 - 2 = 2$
Axis of symmetry: $x = -1$

Answer: $(x + 1)^2 = 8(y - 4)$

Alternate form: Solve for $y$ : $y = \frac{(x + 1)^2}{8} + 4$

3Problem 3hard

❓ Question:

A parabolic satellite dish is $20$ feet wide at the opening and $5$ feet deep. If we place the vertex at the origin with the parabola opening upward, find the equation and determine where the receiver (focus) should be placed.

💡 Show Solution

Set up coordinate system:

Place vertex at origin $(0, 0)$ , parabola opens upward.

Step 1: Identify a point on the parabola:

Width at opening: $20$ feet → extends $10$ feet on each side
Depth: $5$ feet
Points on rim: $(10, 5)$ and $(-10, 5)$

Step 2: Use standard form with vertex at origin: $x^2 = 4py$

Step 3: Substitute point $(10, 5)$ : $10^2 = 4p(5)$ $100 = 20p$ $p = 5$

Step 4: Write the equation: $x^2 = 4(5)y$ $x^2 = 20y$

Or: $y = \frac{x^2}{20}$

Step 5: Find focus location: Focus is at $(0, p) = (0, 5)$

Answers:

Equation: $x^2 = 20y$ or $y = \frac{x^2}{20}$
Receiver (focus) location: $5$ feet above the vertex, at the center

Practical meaning: All signals hitting the dish will reflect to the focus point $5$ feet above the bottom!

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