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

Standard Form (Vertical Major Axis)

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

Where:

• $(h, k)$ = center
• $a$ = semi-major axis (larger value)
• $b$ = semi-minor axis (smaller value)
• $c$ = distance from center to focus

Key relationship: $c^2 = a^2 - b^2$

Key Features of Ellipse

For horizontal major axis $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$:

• Center: $(h, k)$
• Vertices (endpoints of major axis): $(h \pm a, k)$
• Co-vertices (endpoints of minor axis): $(h, k \pm b)$
• Foci: $(h \pm c, k)$ where $c^2 = a^2 - b^2$
• Major axis length: $2a$
• Minor axis length: $2b$

For vertical major axis: Swap the roles (vertices on vertical axis)

Eccentricity

$e = \frac{c}{a}, \quad 0 < e < 1$

• $e$ close to $0$: nearly circular
• $e$ close to $1$: very elongated

Hyperbolas

A hyperbola is the set of all points where the difference of distances to two foci is constant.

Standard Form (Horizontal Transverse Axis)

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

Opens left and right.

Standard Form (Vertical Transverse Axis)

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

Opens up and down.

Where:

• $(h, k)$ = center
• $a$ = distance from center to vertex
• $b$ = determines spread of branches
• $c$ = distance from center to focus

Key relationship: $c^2 = a^2 + b^2$ (note: plus, not minus!)

Key Features of Hyperbola

For horizontal transverse axis $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$:

• Center: $(h, k)$
• Vertices: $(h \pm a, k)$
• Foci: $(h \pm c, k)$ where $c^2 = a^2 + b^2$
• Asymptotes: $y - k = \pm\frac{b}{a}(x - h)$

For vertical transverse axis: Vertices and foci on vertical axis, asymptotes: $y - k = \pm\frac{a}{b}(x - h)$

Asymptotes

The branches of a hyperbola approach (but never touch) the asymptotes.

Rectangle method:

1. Draw rectangle with vertices at $(h \pm a, k \pm b)$
2. Draw diagonals of rectangle
3. These diagonals are the asymptotes

Identifying Conic Sections

From the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ (with $B = 0$):

• Circle: $A = C$ (and same sign)
• Ellipse: $A \neq C$ but same sign
• Parabola: Either $A = 0$ or $C = 0$ (but not both)
• Hyperbola: $A$ and $C$ have opposite signs

Summary Table

Step 2: Find vertices (endpoints of major axis): Vertices are $a$ units left and right of center: $(2 \pm 5, -1) = (7, -1) \text{ and } (-3, -1)$

Step 3: Find co-vertices (endpoints of minor axis): Co-vertices are $b$ units up and down from center: $(2, -1 \pm 3) = (2, 2) \text{ and } (2, -4)$

Step 4: Find foci using $c^2 = a^2 - b^2$: $c^2 = 25 - 9 = 16$ $c = 4$

Foci are $c$ units left and right of center: $(2 \pm 4, -1) = (6, -1) \text{ and } (-2, -1)$

Answers:

• Center: $(2, -1)$
• Vertices: $(7, -1)$ and $(-3, -1)$
• Co-vertices: $(2, 2)$ and $(2, -4)$
• Foci: $(6, -1)$ and $(-2, -1)$
• Major axis length: $2a = 10$
• Minor axis length: $2b = 6$

Step 1: Analyze given information:

• Center: $(0, 0)$
• Focus: $(0, 3)$ → on $y$-axis → vertical major axis
• Vertex: $(0, 5)$ → also on $y$-axis ✓

Step 2: Find $a$ and $c$:

• Distance from center to vertex: $a = 5$
• Distance from center to focus: $c = 3$

Step 3: Use $c^2 = a^2 - b^2$ to find $b$: $3^2 = 5^2 - b^2$ $9 = 25 - b^2$ $b^2 = 16$ $b = 4$

Step 4: Write equation (vertical major axis with center at origin): $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ $\frac{x^2}{16} + \frac{y^2}{25} = 1$

Step 5: Verify:

• Center: $(0, 0)$ ✓
• $a = 5$, so vertices: $(0, \pm 5)$ ✓
• $c = 3$, so foci: $(0, \pm 3)$ ✓
• $b = 4$, so co-vertices: $(\pm 4, 0)$

Answer: $\frac{x^2}{16} + \frac{y^2}{25} = 1$