https://file.aiquickdraw.com/mathbot/file/1728834260317uxjgqoed.jpg

Let's solve each part of the problem step by step:

### Part (a): Finding the coordinates of A and B

1. **Intersection of \( y = x \) with the circle \( x^2 + y^2 = 1 \):**
   - Substitute \( y = x \) into the equation of the circle:
     \[
     x^2 + x^2 = 1
     \]
     \[
     2x^2 = 1
     \]
     \[
     x^2 = \frac{1}{2} \implies x = \frac{1}{\sqrt{2}} \text{ or } x = -\frac{1}{\sqrt{2}}
     \]
   - Since we're interested in the positive intersection (as shown in the diagram), we choose \( x = \frac{1}{\sqrt{2}} \). Then, \( y = \frac{1}{\sqrt{2}} \).
   - So, the coordinates of \( A \) are:
     \[
     A = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)
     \]

2. **Intersection of \( y = x \) with the parabola \( y = x^2 \):**
   - Substitute \( y = x \) into \( y = x^2 \):
     \[
     x = x^2
     \]
     \[
     x^2 - x = 0
     \]
     \[
     x(x - 1) = 0
     \]
   - This gives \( x = 0 \) or \( x = 1 \). Since we are looking for the non-zero intersection (as shown in the diagram), we choose \( x = 1 \). Then, \( y = 1 \).
   - So, the coordinates of \( B \) are:
     \[
     B = (1, 1)
     \]

### Part (b): Finding the area of triangles \( OAX \) and \( OBY \)

1. **Triangle \( OAX \):**
   - The vertices are \( O = (0, 0) \), \( A = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \), and \( X = \left(\frac{1}{\sqrt{2}}, 0\right) \).
   - The base of triangle \( OAX \) is \( \frac{1}{\sqrt{2}} \) (the horizontal distance from \( O \) to \( X \)).
   - The height of triangle \( OAX \) is \( \frac{1}{\sqrt{2}} \) (the vertical distance from \( X \) to \( A \)).
   - The area of \( OAX \) is:
     \[
     \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}} = \frac{1}{4}
     \]

2. **Triangle \( OBY \):**
   - The vertices are \( O = (0, 0) \), \( B = (1, 1) \), and \( Y = (1, 0) \).
   - The base of triangle \( OBY \) is \( 1 \) (the horizontal distance from \( O \) to \( Y \)).
   - The height of triangle \( OBY \) is \( 1 \) (the vertical distance from \( Y \) to \( B \)).
   - The area of \( OBY \) is:
     \[
     \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}
     \]

### Summary of Answers
- **Coordinates**:
  - \( A = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \)
  - \( B = (1, 1) \)
- **Areas**:
  - Area of \( \triangle OAX = \frac{1}{4} \)
  - Area of \( \triangle OBY = \frac{1}{2} \)

Would you like further details or explanations on any part of the solution?

### Follow-up Questions:
1. How would the coordinates of \( A \) and \( B \) change if the line was \( y = -x \)?
2. Can you verify the calculation of the areas using a different method, such as using vectors?
3. What would be the area of the triangle if the parabola were \( y = x^3 \) instead?
4. How would the solution change if the circle's equation was \( x^2 + y^2 = 4 \)?
5. What if the line intersected the parabola at a different angle (e.g., \( y = 2x \))?

### Tip:
Remember that when working with intersections of curves, substituting one equation into another is a common method to find points of intersection. Always check which solution fits the context (e.g., positive or negative intersections).

The circle x^2 + y^2 = 1, the parabola y = x^2 and the line y = x are drawn on the same axes. Let A and B be the points of intersection of y = x with the circle and parabola respectively. XA and YB are drawn perpendicular to the x-axis. Find the coordinates of A and B. Find the area of triangles OAX and OBY.

Solve the problem of finding the intersection points of a circle (x^2 + y^2 = 1), a parabola (y = x^2), and a line (y = x). Learn how to find the coordinates of points A and B, and calculate the areas of triangles OAX and OBY using coordinate geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation