Given Δ𝑃𝑄𝑅 with 𝑃(4;3), 𝑄(3;−2) and R(-2;0). The equation of line 𝑃𝑄 is 𝑦=5𝑥+17 and D is the 𝑥 intercept of line 𝑃𝑄. 𝑀(𝑥;𝑦) is midpoint of PR.If 𝑅 is the centre of a circle with radius 𝑄𝑅, determine the equation of the circle in the form 𝑥2+𝑚𝑥+𝑦2+𝑛𝑦=𝑟2

To find the equation of the circle, we need to identify the center and the radius of the circle. Here's the breakdown of the given information and how we'll approach the solution:

1. **Center of the Circle**:  
   The center of the circle is point \( R(-2, 0) \).

2. **Radius of the Circle**:  
   The radius of the circle is the distance between points \( Q(3, -2) \) and \( R(-2, 0) \).

3. **Equation of the Circle**:  
   The general form of the equation of a circle with center \( (h, k) \) and radius \( r \) is:  
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   Substituting the center \( R(-2, 0) \), the equation becomes:  
   \[
   (x + 2)^2 + (y - 0)^2 = r^2
   \]

### Step-by-Step Solution:

#### 1. Find the Radius \( r \):

The radius \( r \) is the distance between points \( Q(3, -2) \) and \( R(-2, 0) \).  
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:  
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the coordinates of points \( Q \) and \( R \):  
\[
r = \sqrt{(-2 - 3)^2 + (0 - (-2))^2}
\]

#### 2. Expand and Simplify to Find \( r^2 \):

Let's compute the value of \( r^2 \).The value of \( r^2 \) is 29. Thus, the radius \( r \) of the circle is \( \sqrt{29} \).

#### 3. Write the Equation of the Circle:

Substitute the center \( R(-2, 0) \) and \( r^2 = 29 \) into the standard equation of a circle:
\[
(x + 2)^2 + y^2 = 29
\]

#### 4. Expand to the Desired Form:

Now, expand and rewrite the equation in the form \( x^2 + mx + y^2 + ny = r^2 \).

\[
(x + 2)^2 + y^2 = 29 \implies x^2 + 4x + 4 + y^2 = 29
\]

Simplify to:

\[
x^2 + y^2 + 4x = 25
\]

Here, the equation is in the form \( x^2 + mx + y^2 + ny = r^2 \) with:
- \( m = 4 \)
- \( n = 0 \)
- \( r^2 = 25 \)

### Final Answer:

The equation of the circle is:

\[
x^2 + 4x + y^2 = 25
\]

Would you like more details or have any questions about the steps? Here are some related questions you might find interesting:

1. How do you find the equation of a circle given two points on the circle?
2. What is the significance of the midpoint formula in geometry?
3. How do you determine if a point lies on a given circle?
4. How do you derive the distance formula from the Pythagorean theorem?
5. What are some real-life applications of circles in mathematics?

**Tip:** Always double-check your calculations when using the distance formula to ensure accuracy.

Learn how to determine the equation of a circle with a given center and radius using points in coordinate geometry. This problem involves finding the center and radius of the circle using the distance formula and equation of a circle formula. Suitable for high school students studying coordinate geometry.

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