Identify the ellipse characteristics:

Step 1: Determine orientation:

• Form: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$
• $a^2 = 25 > b^2 = 9$ → horizontal major axis
• $a = 5$, $b = 3$
• Center: $(h, k) = (2, -1)$

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: $2b = 6$

Identify the hyperbola characteristics:

Step 1: Determine orientation:

• $y$ term is positive → vertical transverse axis (opens up and down)
• $a^2 = 16$, so $a = 4$
• $b^2 = 9$, so $b = 3$
• Center: $(h, k) = (1, -3)$

Step 2: Find vertices: Vertices are $a$ units above and below center: $(1, -3 \pm 4) = (1, 1) \text{ and } (1, -7)$

Step 3: Find foci using $c^2 = a^2 + b^2$: $c^2 = 16 + 9 = 25$ $c = 5$

Foci are $c$ units above and below center: $(1, -3 \pm 5) = (1, 2) \text{ and } (1, -8)$

Step 4: Find asymptotes: For vertical hyperbola: $y - k = \pm\frac{a}{b}(x - h)$ $y - (-3) = \pm\frac{4}{3}(x - 1)$ $y + 3 = \pm\frac{4}{3}(x - 1)$

Two asymptotes: $y = \frac{4}{3}(x - 1) - 3 = \frac{4}{3}x - \frac{13}{3}$ $y = -\frac{4}{3}(x - 1) - 3 = -\frac{4}{3}x - \frac{5}{3}$

Answers:

• Center: $(1, -3)$
• Vertices: $(1, 1)$ and $(1, -7)$
• Foci: $(1, 2)$ and $(1, -8)$
• Asymptotes: $y = \frac{4}{3}x - \frac{13}{3}$ and $y = -\frac{4}{3}x - \frac{5}{3}$

Determine ellipse characteristics:

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

Step 1: Identify the center: Standard form: (x - h)²/a² + (y - k)²/b² = 1 Center: (h, k) = (1, -2)

Step 2: Identify a² and b²: a² = 25, so a = 5 b² = 9, so b = 3

Step 3: Determine major axis: Since a² > b², major axis is horizontal

Step 4: Find vertices (on major axis): Vertices are a units left and right of center (1 - 5, -2) = (-4, -2) (1 + 5, -2) = (6, -2)

Step 5: Find c using c² = a² - b²: c² = 25 - 9 = 16 c = 4

Step 6: Find foci (on major axis): Foci are c units from center along major axis (1 - 4, -2) = (-3, -2) (1 + 4, -2) = (5, -2)

Answer: Center (1, -2), Vertices (-4, -2) and (6, -2), Foci (-3, -2) and (5, -2)

Step 1: Determine orientation: Vertices and foci on x-axis → horizontal hyperbola Standard form: x²/a² - y²/b² = 1

Step 2: Find a from vertices: Vertices at (±a, 0) = (±3, 0) a = 3, so a² = 9

Step 3: Find c from foci: Foci at (±c, 0) = (±5, 0) c = 5

Step 4: Find b using c² = a² + b²: 5² = 3² + b² 25 = 9 + b² b² = 16 b = 4

Step 5: Write the equation: x²/9 - y²/16 = 1

Step 6: Find asymptotes (for reference): y = ±(b/a)x = ±(4/3)x

Answer: x²/9 - y²/16 = 1