Let's go through each question step by step.

Question 3: Find the center of the circle for the equation:

$x^2 + y^2 - 16x - 8y + 31 = 0$

This is a general form of a circle equation:
$x^2 + y^2 + Dx + Ey + F = 0$

To find the center, we need to convert this into the standard form of a circle equation: $(x - h)^2 + (y - k)^2 = r^2$ Where $(h, k)$ is the center.

Step 1: Completing the square

• Group the $x$-terms and $y$-terms: $(x^2 - 16x) + (y^2 - 8y) = -31$

• Complete the square for both: $x^2 - 16x \quad \text{becomes} \quad (x - 8)^2 - 64$ $y^2 - 8y \quad \text{becomes} \quad (y - 4)^2 - 16$

• Substitute back: $(x - 8)^2 - 64 + (y - 4)^2 - 16 = -31$

• Simplify: $(x - 8)^2 + (y - 4)^2 = 49$

Thus, the center is $(8, 4)$, and the radius is 7.

Answer for Question 3:

c. (8, 4)

Question 4: Find the standard equation of a circle with center (-4,3) and radius $\sqrt{7}$

The general form of a circle equation is: $(x - h)^2 + (y - k)^2 = r^2$ Where $(h, k)$ is the center and $r$ is the radius.

Given:

• Center: $(-4, 3)$
• Radius: $\sqrt{7}$

Substitute these values: $(x + 4)^2 + (y - 3)^2 = (\sqrt{7})^2 = 7$

Answer for Question 4:

c. (x + 4)^2 + (y - 3)^2 = 7

Question 5: Determine the vertex, focus, directrix, and axis of symmetry of the following parabolas:

(1) $x^2 = -4y$

This is a vertical parabola in the form of: $x^2 = 4py$

• The vertex is at (0, 0).
• Since $4p = -4$, we get $p = -1$. The focus is at $(0, -1)$.
• The directrix is $y = 1$.
• The axis of symmetry is the $y$-axis, $x = 0$.

(2) $3y^2 = 24x$

This is a horizontal parabola in the form of: $y^2 = 4px$

• Divide both sides by 3:
  $y^2 = 8x \implies 4p = 8 \implies p = 2$
• The vertex is at (0, 0).
• The focus is at $(2, 0)$.
• The directrix is $x = -2$.
• The axis of symmetry is the $x$-axis, $y = 0$.

(3) $(y + \frac{5}{2})^2 = -5(x - \frac{2}{9})$

This is a horizontal parabola of the form: $(y - k)^2 = 4p(x - h)$

• The vertex is at $\left( \frac{2}{9}, -\frac{5}{2} \right)$.
• Since $4p = -5$, $p = -\frac{5}{4}$, so the focus is at $\left( \frac{2}{9} - \frac{5}{4}, -\frac{5}{2} \right) = \left( -\frac{13}{36}, -\frac{5}{2} \right)$.
• The directrix is $x = \frac{2}{9} + \frac{5}{4} = \frac{58}{36} = \frac{29}{18}$.
• The axis of symmetry is $y = -\frac{5}{2}$.

Would you like more details on any of these questions?

5 Related Questions:

1. How do you find the radius of a circle given its equation?
2. How does completing the square help in converting a quadratic equation into standard form?
3. What is the geometric meaning of the directrix in a parabola?
4. How can you differentiate between vertical and horizontal parabolas?
5. How do you calculate the focus of a parabola from its equation?

Tip: In any conic section, the value of $p$ helps determine the distance between the vertex and both the focus and the directrix.