Let's solve the problems step-by-step:

Question 9:

The given equation of the ellipse is:

$\frac{(y+1)^2}{25} + \frac{(x+2)^2}{4} = 1$

This is the standard form of an ellipse, where:

• $a^2 = 25$ (larger denominator, semi-major axis squared),
• $b^2 = 4$ (smaller denominator, semi-minor axis squared).

The length of the minor axis is $2b$: $b = \sqrt{4} = 2$ $\text{Length of the minor axis: } 2b = 2 \times 2 = 4$

Answer for Question 9: The length of the minor axis is 4.

Question 10:

The given equation of the hyperbola is:

$\frac{(y-2)^2}{4} - \frac{(x-5)^2}{1} = 1$

This is the standard form of a vertical hyperbola, where:

• Center: $(h, k) = (5, 2)$,
• $a^2 = 4$, so $a = \sqrt{4} = 2$,
• $b^2 = 1$, so $b = \sqrt{1} = 1$,
• The slopes of the asymptotes are $\pm \frac{a}{b} = \pm \frac{2}{1} = \pm 2$.

Analyzing the given statements:

1. The vertices are (5, 0) and (5, 4):

   • For a vertical hyperbola, the vertices are at $(h, k \pm a) = (5, 2 \pm 2) = (5, 0)$ and $(5, 4)$.
   • This is true.

2. The vertices are (4, 2) and (6, 2):

   • These would be correct for a horizontal hyperbola, but this hyperbola is vertical.
   • This is false.

3. The foci are $(5, 5 \pm \sqrt{2})$:

   • The foci are $(h, k \pm c)$, where $c = \sqrt{a^2 + b^2} = \sqrt{4 + 1} = \sqrt{5}$.
   • The correct foci are $(5, 2 \pm \sqrt{5})$, not $(5, 5 \pm \sqrt{2})$.
   • This is false.

4. The slope of the asymptotes is $\pm 2$:

   • As calculated earlier, the slopes of the asymptotes are $\pm 2$.
   • This is true.

5. The slope of the asymptotes is $\pm \frac{1}{2}$:

   • This is incorrect because the slope is $\pm 2$.
   • This is false.

6. The center is (5, 2):

   • From the equation, the center is $(5, 2)$.
   • This is true.

Correct answers for Question 10:

• The vertices are (5, 0) and (5, 4).
• The slope of the asymptotes is $\pm 2$.
• The center is (5, 2).

Would you like further clarification or additional details?

Related Questions:

1. How do you determine whether a hyperbola is horizontal or vertical?
2. What is the difference between the vertices and co-vertices of an ellipse?
3. How is the length of the major axis of an ellipse calculated?
4. Can a hyperbola ever have a slope of asymptotes as zero? Why or why not?
5. What role does $c$ (focal distance) play in the definition of conic sections?

Tip:

To determine the orientation of a hyperbola, always check the sign and order of the squared terms in its standard equation.