Year 11 Maths: Write equations of circles centered at the origin from graphs

Introduction to Circles

Hello everyone! Today, we are going to learn about writing equations of circles that are centered at the origin.

What is a Circle?

A circle is a round shape where every point on the edge is the same distance from the center. In our case, the center of the circle is at the point (0, 0), which is called the origin.

The Equation of a Circle

The general equation for a circle centered at the origin is:

$$x^2 + y^2 = r^2$$

• r is the radius of the circle.
• (x, y) are the coordinates of any point on the circle.

Understanding the Radius

The radius is the distance from the center of the circle to any point on the circle. To find the radius from a graph, look at how far the circle goes from the center to the edge.

Example 1

Imagine you have a circle that reaches out to the point (3, 0). The radius is 3 units. So, we can write the equation:

$$x^2 + y^2 = 3^2$$

This simplifies to:

$$x^2 + y^2 = 9$$

Example 2

If you have a circle that reaches to the point (4, 0), the radius is 4 units. The equation would be:

$$x^2 + y^2 = 4^2$$

Which simplifies to:

$$x^2 + y^2 = 16$$

Key Rules

1. Identify the Radius: Look at the graph and find the distance from the origin to the edge of the circle.
2. Plug in the Radius: Use the radius in the equation (x^2 + y^2 = r^2).
3. Simplify: Make sure to square the radius.

Tips and Tricks

• Remember, the radius will always be a positive number.
• If the circle touches the x-axis or y-axis, check those points to find the radius.
• Always double-check your graph to ensure you have the correct radius.

Questions

Easy Level Questions

1. Write the equation of a circle with a radius of 1.
2. Write the equation of a circle with a radius of 2.
3. Write the equation of a circle with a radius of 3.
4. Write the equation of a circle with a radius of 4.
5. Write the equation of a circle with a radius of 5.
6. Write the equation of a circle with a radius of 6.
7. Write the equation of a circle with a radius of 7.
8. Write the equation of a circle with a radius of 8.
9. Write the equation of a circle with a radius of 9.
10. Write the equation of a circle with a radius of 10.

Medium Level Questions

11. Write the equation of a circle that passes through the point (5, 0).
12. Write the equation of a circle that passes through the point (6, 0).
13. Write the equation of a circle that passes through the point (7, 0).
14. Write the equation of a circle that passes through the point (8, 0).
15. Write the equation of a circle that passes through the point (9, 0).
16. Write the equation of a circle that passes through the point (10, 0).
17. Write the equation of a circle that passes through the point (3, 4).
18. Write the equation of a circle that passes through the point (0, 5).
19. Write the equation of a circle that passes through the point (-4, 0).
20. Write the equation of a circle that passes through the point (0, -3).

Hard Level Questions

21. Write the equation of a circle that has a radius of √5.
22. Write the equation of a circle that has a radius of √10.
23. Write the equation of a circle that has a radius of √15.
24. Write the equation of a circle that has a radius of √20.
25. Write the equation of a circle that has a radius of √30.
26. Write the equation of a circle that has a radius of √50.
27. Write the equation of a circle that passes through the point (√2, √2).
28. Write the equation of a circle that passes through the point (√3, √3).
29. Write the equation of a circle that passes through the point (-√2, √2).
30. Write the equation of a circle that passes through the point (√5, -√5).

Answers and Explanations

Easy Level Answers

1. $$x^2 + y^2 = 1$$
2. $$x^2 + y^2 = 4$$
3. $$x^2 + y^2 = 9$$
4. $$x^2 + y^2 = 16$$
5. $$x^2 + y^2 = 25$$
6. $$x^2 + y^2 = 36$$
7. $$x^2 + y^2 = 49$$
8. $$x^2 + y^2 = 64$$
9. $$x^2 + y^2 = 81$$
10. $$x^2 + y^2 = 100$$

Medium Level Answers

11. $$x^2 + y^2 = 25$$ (radius is 5)
12. $$x^2 + y^2 = 36$$ (radius is 6)
13. $$x^2 + y^2 = 49$$ (radius is 7)
14. $$x^2 + y^2 = 64$$ (radius is 8)
15. $$x^2 + y^2 = 81$$ (radius is 9)
16. $$x^2 + y^2 = 100$$ (radius is 10)
17. $$x^2 + y^2 = 25$$ (radius is 5)
18. $$x^2 + y^2 = 25$$ (radius is 5)
19. $$x^2 + y^2 = 16$$ (radius is 4)
20. $$x^2 + y^2 = 9$$ (radius is 3)

Hard Level Answers

21. $$x^2 + y^2 = 5$$
22. $$x^2 + y^2 = 10$$
23. $$x^2 + y^2 = 15$$
24. $$x^2 + y^2 = 20$$
25. $$x^2 + y^2 = 30$$
26. $$x^2 + y^2 = 50$$
27. $$x^2 + y^2 = 4$$ (radius is 2)
28. $$x^2 + y^2 = 6$$ (radius is √6)
29. $$x^2 + y^2 = 8$$ (radius is 2√2)
30. $$x^2 + y^2 = 10$$ (radius is √10)

I hope this guide helps you understand how to write equations of circles centered at the origin! Keep practicing, and you’ll get the hang of it in no time!