Circles · Parabolas · Ellipses · Hyperbolas — all four, concept first
MAT 172 — Final Prep
All four conics come from the same source: slice a double cone at different angles. Slice straight across → circle. Slight tilt → ellipse. Parallel to the cone's side → parabola. Through both cones → hyperbola. The equation tells you which slice you're looking at.
ConceptA circle is every point exactly the same distance (radius) from a center point. That's it. The equation is just the Pythagorean theorem: the distance from (x,y) to (h,k) equals r.
(x − h)² + (y − k)² = r²
Center: (h, k) · Radius: r · All points equidistant from center
Interactive circle — drag the sliders
0
0
3
Equation
x²+y²=9
Circumference
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Area
—
Standard form → center & radius
Find the center and radius of x² + y² − 6x + 4y − 3 = 0
1
Group x and y terms: (x² − 6x) + (y² + 4y) = 3
2
Complete the square for x: take half of −6 = −3, square it = 9. Add to both sides: (x² − 6x + 9) + (y² + 4y) = 3 + 9
3
Complete the square for y: take half of 4 = 2, square it = 4. Add to both sides: (x − 3)² + (y + 2)² = 3 + 9 + 4 = 16
4
Standard form: (x − 3)² + (y + 2)² = 16. So r² = 16 → r = 4.
Center: (3, −2) · Radius: 4
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Completing the square — step by step
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For x² + bx: take half of b, square it, add to both sides.
Example: x² − 6x → half of −6 = −3 → (−3)² = 9 → write (x − 3)² and add 9 to the right side.
The most common mistake: forgetting to add the same number to both sides. If you add 9 to the left in the parentheses, you must add 9 to the right.
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General form → standard form (the full process)
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Ax² + Ay² + Cx + Dy + E = 0 (when A = 1):
1. Move E to the right side
2. Group: (x² + Cx) + (y² + Dy) = −E
3. Complete the square for each group
4. Factor into (x − h)² + (y − k)² = r²
If r² comes out negative → no real circle exists. If r² = 0 → a single point.
Circle vocabulary
Center (h, k)
The fixed point equidistant from all points on the circle.
Radius r
Distance from center to any point on the circle. Always positive.
Diameter
2r — the longest chord through the center.
Standard form
(x−h)²+(y−k)²=r²
General form
x²+y²+Cx+Dy+E=0. Use completing the square to convert to standard.
Exam reminders
The equation is r², not r. If the right side is 25, then r = 5 not 25.
h and k are subtracted in standard form. (x+3)² means h = −3.
If A ≠ 1 in general form, divide everything by A first.
Unit circle: center (0,0), r = 1. Equation: x² + y² = 1.
Completing the square: always add the same value to BOTH sides.
ConceptA parabola is every point equidistant from a fixed point (the focus) AND a fixed line (the directrix). The result: a U-shaped curve where the vertex is halfway between focus and directrix. Every satellite dish, headlight reflector, and projectile path is a parabola.
Vertical: y = a(x−h)² + k · Horizontal: x = a(y−k)² + h
Vertex: (h, k) · Opens up/down if vertical, left/right if horizontal · a determines width and direction
Interactive parabola
0
0
1
Equation
y=x²
Opens
Up
Axis of sym.
x=0
Finding vertex form from standard form
Write y = 2x² − 8x + 5 in vertex form
1
Factor out a from x terms: y = 2(x² − 4x) + 5
2
Complete the square inside: half of −4 = −2, squared = 4. Add and subtract: y = 2(x² − 4x + 4 − 4) + 5
Vertex form: y = 2(x−2)² − 3 · Vertex: (2, −3) · Opens up (a > 0)
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Focus and directrix — what they actually are
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For y = a(x−h)² + k, the focus is at (h, k + 1/(4a)) and the directrix is y = k − 1/(4a).
The focus is INSIDE the parabola. The directrix is a line OUTSIDE it on the opposite side. Every point on the parabola is equidistant from both.
Real-world connection: Satellite dishes are parabolas with a receiver at the focus. Parallel signals from space hit the dish and all reflect to the same focus point — that's why the geometry works. Your headlights use the same principle in reverse: a bulb at the focus sends light out in parallel beams.
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Vertical vs. horizontal parabolas
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Vertical: y = a(x−h)² + k. The x is squared. Opens up (a > 0) or down (a < 0). Axis of symmetry is vertical: x = h.
Horizontal: x = a(y−k)² + h. The y is squared. Opens right (a > 0) or left (a < 0). Axis of symmetry is horizontal: y = k.
Quick check: which variable is squared? That variable is the one that fans out. The other variable is the axis direction.
Parabola vocabulary
Vertex (h, k)
The turning point. Maximum if a < 0, minimum if a > 0.
Focus
Fixed point inside the parabola. Distance from focus = distance to directrix for every point.
Directrix
Fixed line outside the parabola, opposite the focus.
Axis of symmetry
The line through the vertex and focus that cuts the parabola in half.
Latus rectum
Chord through the focus parallel to the directrix. Length = |1/a|.
Exam reminders
a > 0 → opens up or right. a < 0 → opens down or left.
|a| large → narrow parabola. |a| small → wide parabola.
Vertex is always the min or max — use it for optimization problems.
Which variable is squared? That's the one that opens. x² → vertical, y² → horizontal.
Completing the square: when a ≠ 1, factor it out FIRST before completing.
ConceptAn ellipse is every point where the SUM of distances to two fixed points (foci) is constant. Stretch a circle and you get an ellipse. The longer axis is called major, the shorter is minor. Planets orbit in ellipses — the sun is at one focus, not the center.
The larger denominator (a²) tells you which axis is major.
• If a² is under x²: major axis is horizontal → ellipse is wider than tall.
• If a² is under y²: major axis is vertical → ellipse is taller than wide.
Foci are always on the major axis — they follow the larger denominator.
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The sum of distances definition (why it works)
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For any point P on the ellipse: d(P, F₁) + d(P, F₂) = 2a (always constant).
At the vertex: one distance is a+c and the other is a−c. They add to 2a ✓
Garden ellipse trick: Stick two stakes in the ground (the foci), tie a string of length 2a between them, pull it taut with a pencil, and trace — the pencil draws a perfect ellipse because the string length stays constant.
Ellipse vocabulary
Semi-major axis a
Half the longest diameter. Always a > b. Vertices are a units from center.
Semi-minor axis b
Half the shortest diameter. Co-vertices are b units from center.
Foci (plural of focus)
Two fixed points inside the ellipse. c units from center along major axis.
c² = a² − b²
The relationship between a, b, and c. Always memorize this.
Eccentricity e = c/a
How "stretched" the ellipse is. e = 0 is a circle, e close to 1 is very elongated.
Exam reminders
a > b ALWAYS. If b > a in the problem, swap them.
c² = a² − b² (not + like Pythagorean theorem). The foci are INSIDE.
Foci follow the major axis — same variable as the larger denominator.
Sum of distances from any point to both foci = 2a.
Circle is a special ellipse where a = b and c = 0 (foci merge at center).
ConceptA hyperbola is every point where the DIFFERENCE of distances to two foci is constant. Unlike an ellipse (sum), a hyperbola has two separate branches. The asymptotes are the lines the branches approach but never touch — they're your guide for graphing.
The central rectangle method — how to draw asymptotes
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1. Draw a rectangle centered at (h, k) with width 2a and height 2b.
2. Draw the diagonals of that rectangle — those ARE the asymptotes.
3. The hyperbola branches open from the vertices and hug the asymptotes.
For horizontal hyperbola: asymptotes are y − k = ±(b/a)(x − h) For vertical hyperbola: asymptotes are y − k = ±(a/b)(x − h)
Notice: vertical swaps a and b in the asymptote slope. Easy to mess up — always check which term is positive.
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Ellipse vs. Hyperbola — the critical differences
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Feature
Ellipse
Hyperbola
Sign between terms
+ (plus)
− (minus)
c² formula
c² = a² − b²
c² = a² + b²
Branches
One closed curve
Two open branches
Eccentricity
0 < e < 1
e > 1
Asymptotes
None
Yes — y = ±(b/a)x
Hyperbola vocabulary
Vertices
The two points where the branches are closest together. Distance a from center.
Foci
c units from center, beyond the vertices. c² = a² + b² (note the +, not −).
Asymptotes
Lines the branches approach but never touch. Slope = ±b/a (horizontal) or ±a/b (vertical).
Transverse axis
The axis containing both vertices and foci. Length = 2a.
Conjugate axis
Perpendicular to transverse, through center. Length = 2b.
Exam reminders
The POSITIVE term tells you which direction the branches open.
c² = a² + b² — foci are OUTSIDE the hyperbola (beyond the vertices).
Asymptote slope for horizontal: b/a. For vertical: a/b. Don't mix them up.
Eccentricity e = c/a. For hyperbola, e is ALWAYS greater than 1.
Difference of distances = 2a (the constant). |d₁ − d₂| = 2a.