Consider the graph G of the function f (x) = x 2 and the point M(5,2) on the Cartesian plane. (a) Find the coordinates of a point A on G and a point B on x-axis, given that M(5,2) is the midpoint of the line segment [AB]. (b) Find the coordinates of a point C on G, given that the distance between M and C is 23.

Let's solve each part of the problem step-by-step.

### Problem Overview:
We are given the function \( f(x) = x^2 \) and the point \( M(5, 2) \). We need to find the points \( A \) and \( B \) under given conditions.

### (a) Finding the coordinates of a point \( A \) on \( G \) (where \( G \) is the graph of \( f(x) = x^2 \)) and a point \( B \) on the x-axis, given that \( M(5, 2) \) is the midpoint of the line segment \( [AB] \).

1. **Midpoint Formula**: 
   The midpoint \( M \) of the line segment \( [AB] \) is given by:

   \[
   M = \left(\frac{a_1 + b_1}{2}, \frac{a_2 + b_2}{2}\right)
   \]

   Given \( M = (5, 2) \), we have:

   \[
   \frac{a_1 + b_1}{2} = 5 \quad \text{and} \quad \frac{a_2 + b_2}{2} = 2
   \]

   Solving these equations:

   \[
   a_1 + b_1 = 10 \quad \text{(Equation 1)}
   \]

   \[
   a_2 + b_2 = 4 \quad \text{(Equation 2)}
   \]

2. **Substituting Line Equations**: 
   Since \( A \) is on \( G: y = x^2 \), let \( A = (a_1, a_1^2) \). Since \( B \) is on the x-axis, \( B = (b_1, 0) \).

   Substitute these into the equations derived from the midpoint:

   From Equation 1:

   \[
   a_1 + b_1 = 10
   \]

   From Equation 2:

   \[
   a_1^2 + 0 = 4
   \]

   Simplify:

   \[
   a_1^2 = 4
   \]

   So, \( a_1 = 2 \) or \( a_1 = -2 \).

3. **Find Corresponding \( b_1 \)**:
   Using \( a_1 = 2 \):

   \[
   2 + b_1 = 10 \implies b_1 = 8
   \]

   So, one pair is \( A = (2, 4) \) and \( B = (8, 0) \).

   Using \( a_1 = -2 \):

   \[
   -2 + b_1 = 10 \implies b_1 = 12
   \]

   So, the other pair is \( A = (-2, 4) \) and \( B = (12, 0) \).

### Final Answers for (a):
- Coordinates of \( A = (2, 4) \) and \( B = (8, 0) \).
- Or, coordinates of \( A = (-2, 4) \) and \( B = (12, 0) \).

### (b) Finding the coordinates of a point \( C \) on \( G \), given that the distance between \( M \) and \( C \) is 23.

1. **Distance Formula**:
   The distance between points \( M(5, 2) \) and \( C(c_1, c_2) \) is given by:

   \[
   \sqrt{(c_1 - 5)^2 + (c_2 - 2)^2} = 23
   \]

   Since \( C \) is on \( G \), we have \( c_2 = c_1^2 \). Substituting this into the distance formula:

   \[
   \sqrt{(c_1 - 5)^2 + (c_1^2 - 2)^2} = 23
   \]

   Squaring both sides:

   \[
   (c_1 - 5)^2 + (c_1^2 - 2)^2 = 529
   \]

   Expanding each term:

   \[
   (c_1 - 5)^2 = c_1^2 - 10c_1 + 25
   \]

   \[
   (c_1^2 - 2)^2 = c_1^4 - 4c_1^2 + 4
   \]

   Substitute these into the equation:

   \[
   c_1^2 - 10c_1 + 25 + c_1^4 - 4c_1^2 + 4 = 529
   \]

   Simplify:

   \[
   c_1^4 - 3c_1^2 - 10c_1 + 29 = 529
   \]

   \[
   c_1^4 - 3c_1^2 - 10c_1 - 500 = 0
   \]

2. **Solve the Quartic Equation**:
   This quartic equation is complex and requires numerical methods to find exact values of \( c_1 \). However, it can be approximated or solved using specialized software or iterative methods.

### Summary of Solutions:
- For (a), \( A = (2, 4) \), \( B = (8, 0) \) or \( A = (-2, 4) \), \( B = (12, 0) \).
- For (b), finding \( C \) requires solving the quartic equation numerically.

Would you like further clarification or details on solving the quartic equation?

### Relative Questions:
1. How do you solve quartic equations analytically?
2. What methods can be used to approximate solutions of complex equations?
3. How can you verify if a point lies on a given function graph?
4. What is the significance of distance and midpoint calculations in coordinate geometry?
5. What are numerical methods used for solving polynomial equations?

### Tip:
When dealing with higher-degree equations, graphing or numerical approximation methods can often provide insights into possible solutions quickly.

Explore a coordinate geometry problem involving a quadratic function and the midpoint formula. Solve for points A and B on the function f(x) = x^2 and the x-axis, given that M(5, 2) is the midpoint of [AB]. Also, find the coordinates of point C on the function such that the distance from M(5, 2) to C is 23. Suitable for Grades 10-12.

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