Conic Sections

A2.I.1Circles01/04

Circles — standard form.

Step01of 04

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

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

Reads as "every point $(x, y)$ is distance $r$ from the center" — the distance formula squared.

A2.I.1 — Circles — standard form.

Step 01 of 04

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

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

Reads as "every point $(x, y)$ is distance $r$ from the center" — the distance formula squared.

Step 02 of 04

Read the center and radius off the equation.

$(x - 4)^{2} + (y + 1)^{2} = 16$	→center $(4, - 1)$ , radius $4$
$x^{2} + y^{2} = 9$	→center $(0, 0)$ , radius $3$

Sign trap on h and k: equation reads MINUS, so $y + 1 = y - (- 1)$ means $k = - 1$ . Radius is $r^{2}$ , not $r^{2}$ directly.

Step 03 of 04

From general form — complete the square in both variables.

Given $x^{2} + y^{2} - 6 x + 4 y - 12 = 0$ :

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

(x - 3)^{2} + (y + 2)^{2} = 25

Center $(3, - 2)$ , radius $5$ .

Step 04 of 04

Write the equation given other info.

Given center + radius	→plug into the standard form
Given center + point on circle	→distance formula gives r, then plug in
Given two endpoints of diameter	→midpoint = center; half the distance = radius

Key insight

Standard form bakes the center and radius into the equation. Sign flip when reading h and k (it's "minus" inside); radius is

of the right side. Complete the square in both variables to get from general form.

A2.I.2 — Parabolas — focus and directrix.

Step 01 of 04

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex sits exactly halfway between them.

Vertical (opens up/down)	→ $(x - h)^{2} = 4 p (y - k)$
Horizontal (opens left/right)	→ $(y - k)^{2} = 4 p (x - h)$

$p$ is the distance from vertex to focus (and from vertex to directrix). Sign of $p$ tells you direction: $p > 0$ opens up/right, $p < 0$ opens down/left.

Step 02 of 04

Worked example. A parabola has vertex $(2, - 1)$ and focus $(2, 3)$ .

Vertex and focus share the same x-coordinate → opens vertically. Focus is ABOVE vertex (y goes up by 4) → opens up. $p = 4$ .

(x - 2)^{2} = 4 (4) (y + 1) = 16 (y + 1)

Directrix: line $y = - 1 - 4 = - 5$ (same distance below vertex as focus is above).

Step 03 of 04

From the equation, read everything.

Given $(x + 1)^{2} = 8 (y - 3)$ :

Vertex	→ $(- 1, 3)$
4p	→8 → p = 2
Opens	→vertically (the y has the linear term), up (p > 0)
Focus	→ $(- 1, 3 + 2) = (- 1, 5)$
Directrix	→ $y = 3 - 2 = 1$

Step 04 of 04

Vertical vs. horizontal — which variable is squared?

$(x - h)^{2} = 4 p (y - k)$	→opens up or down — vertical axis of symmetry
$(y - k)^{2} = 4 p (x - h)$	→opens left or right — horizontal axis of symmetry

The squared variable is the one PERPENDICULAR to the opening direction.

Key insight

(x - h)^{2} = 4 p (y - k)

opens vertically; swap roles for horizontal. Sign of

p

= direction; magnitude of

p

= distance from vertex to focus and to directrix.

A2.I.3 — Ellipses.

Step 01 of 04

Standard form (centered at origin).

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

$a$ = semi-major axis (the LONGER one), $b$ = semi-minor axis. By convention, $a > b$ always.

If $a^{2} > b^{2}$ under $x^{2}$	→horizontal ellipse — wider than tall
If $a^{2} > b^{2}$ under $y^{2}$	→vertical ellipse — taller than wide

Step 02 of 04

Vertices and co-vertices are endpoints of the axes.

Vertices	→endpoints of the major axis, distance $a$ from center
Co-vertices	→endpoints of the minor axis, distance $b$ from center
Foci	→two points on the major axis, distance $c$ from center where $c^{2} = a^{2} - b^{2}$

Step 03 of 04

Worked example. Identify everything for $\frac{x ^{2}}{25} + \frac{y ^{2}}{9} = 1$ .

a²	→25 → a = 5
b²	→9 → b = 3
Major axis	→horizontal (larger denominator under x²)
Vertices	→ $(\pm 5, 0)$
Co-vertices	→ $(0, \pm 3)$
c²	→a² − b² = 16 → c = 4
Foci	→ $(\pm 4, 0)$

Step 04 of 04

Translated form. Center at $(h, k)$ :

\frac{( x - h ) ^{2}}{a ^{2}} + \frac{( y - k ) ^{2}}{b ^{2}} = 1

Same logic, just shift everything by $(h, k)$ .

Defining property. An ellipse is the set of points where the SUM of distances to the two foci is constant (equal to $2 a$ ).

Key insight

Ellipse:

x^{2} / a^{2} + y^{2} / b^{2} = 1

with

a > b

. Larger denominator marks the major axis. Foci on major axis at distance

c = a^{2} - b^{2}

from center.

A2.I.4 — Hyperbolas.

Step 01 of 04

Standard form (centered at origin).

Horizontal (opens left/right)	→ $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$
Vertical (opens up/down)	→ $\frac{y ^{2}}{a ^{2}} - \frac{x ^{2}}{b ^{2}} = 1$

The DIFFERENCE (not sum) and the POSITIVE term tell you the orientation — the variable in the positive term is the direction the hyperbola opens.

Step 02 of 04

Vertices and asymptotes.

Vertices	→distance $a$ from center along the opening direction
Foci	→distance $c$ where $c^{2} = a^{2} + b^{2}$ (note: PLUS for hyperbola, MINUS for ellipse)
Asymptotes (horizontal hyperbola)	→ $y = \pm \frac{b}{a} x$
Asymptotes (vertical hyperbola)	→ $y = \pm \frac{a}{b} x$

Step 03 of 04

Worked example. $\frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1$ .

Opens	→horizontally (positive term is x²)
a²	→16 → a = 4
b²	→9 → b = 3
Vertices	→ $(\pm 4, 0)$
c²	→a² + b² = 25 → c = 5
Foci	→ $(\pm 5, 0)$
Asymptotes	→ $y = \pm \frac{3}{4} x$

Step 04 of 04

Defining property. A hyperbola is the set of points where the ABSOLUTE DIFFERENCE of distances to the two foci is constant (equal to $2 a$ ). Compare to ellipse (sum constant).

The four conics in one comparison.

Circle	→one point + radius
Parabola	→distance to focus = distance to directrix
Ellipse	→sum of distances to two foci = constant
Hyperbola	→absolute difference of distances to two foci = constant

Key insight

Hyperbola = MINUS sign in standard form. Open in the direction of the positive term.

c^{2} = a^{2} + b^{2}

(plus, not minus). Two asymptotes, slopes

\pm b / a

(for horizontal) — they bound the branches at infinity.

Free diagnostic

Test your understanding: 10 questions, ~10 min.

Pulled live from our bank for A2.I · scored instantly with worked solutions · free, no sign-up.

Start diagnostic →Print worksheets

A2.I.1 – A2.I.4 · CCSS G.GPE.A.1 / G.GPE.A.2 / G.GPE.A.3·Full curriculum →