What are the coordinates of the center of the circle given by the equation x2+y2-16x-8y+31=0?

Question 3Select one:

a.

(-8,4)

b.

(8,-4)

c.

(8,4)

d.

(-8,-4)

Clear my choice

Question **4**

Not yet answered

Marked out of 1.00

Flag question

Question text

Give the standard equation of the circle satisfying the given condition: center at (-4,3), radius sqrt(7).

Question 4Select one:

a.

(x-4)2 + (y+3)2 = 49

b.

(x-4)2 + (y+3)2 = 7

c.

(x+4)2 + (y-3)2 = 7

d.

(x+4)2 + (y-3)2 = 49

Clear my choice

Question **5**

Not yet answered

Marked out of 1.00

Flag question

Question text

Determine the vertex, focus, directrix, and axis of symmetry of the parabola with the given equations: **Remember to enclose vertices and focus in parentheses, i.e. (8, 5); (5, 10). For directrix and axis of symmetry, put your answer in a complete equation, i.e. y = 4.7; x = 0**

X2 = -4y

3y2 = 24x

(y + 5/2)2 = -5(x - 2/9)

Item

Vertex

Focus

Directrix

Axis of Symmetry

(1)

Answer 1 Question 5

 Answer 2 Question 5

Answer 3 Question 5

 Answer 4 Question 5

(2)

Answer 5 Question 5

Answer 6 Question 5 

Answer 7 Question 5

Answer 8 Question 5 

(3)

Answer 9 Question 5

Answer 10 Question 5 

 Answer 11 Question 5

 Answer 12 Question 5


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.

What are the coordinates of the center of the circle given by the equation x^2 + y^2 - 16x - 8y + 31 = 0?
Give the standard equation of the circle satisfying the given condition: center at (-4,3), radius sqrt(7).
Determine the vertex, focus, directrix, and axis of symmetry of the parabola with the given equations: X^2 = -4y, 3y^2 = 24x, (y + 5/2)^2 = -5(x - 2/9).

This page provides a detailed solution to finding the center of a circle from its equation, determining the standard equation for a circle with a given center and radius, and analyzing the vertex, focus, directrix, and axis of symmetry for parabolas. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation