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

Introduction

Hello students. Today, we will be learning how to write equations of circles centered at the origin from properties. This may sound a bit complicated, but don’t worry, I’ll explain it to you in a simple and clear way, with examples.

Key Rules

A circle is a set of points in a plane that are all the same distance from a fixed point called the centre. When the centre of the circle is at the origin (0,0), the equation of the circle is given by:

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

Where r is the radius of the circle.

This equation tells us that any point (x, y) on the circle is a distance r from the origin.

Tips and Tricks

1. Always make sure that the circle is centred at the origin. If it isn’t, you will need to use a different equation.
2. Try to visualize the circle and its radius. This will help you understand the equation better.
3. Remember, the radius r must always be a positive number.

Example

Let’s say we have a circle centred at the origin and it has a radius of 3 units. The equation of this circle would be:

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

So, the equation of the circle is:

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

Practice Questions

Easy

1. Write the equation of a circle centred at the origin with radius 4.
2. Write the equation of a circle centred at the origin with radius 6.
3. Write the equation of a circle centred at the origin with radius 5.
4. Write the equation of a circle centred at the origin with radius 2.
5. Write the equation of a circle centred at the origin with radius 7.
6. Write the equation of a circle centred at the origin with radius 8.
7. Write the equation of a circle centred at the origin with radius 1.
8. Write the equation of a circle centred at the origin with radius 3.
9. Write the equation of a circle centred at the origin with radius 10.
10. Write the equation of a circle centred at the origin with radius 11.
11. Write the equation of a circle centred at the origin with radius 12.
12. Write the equation of a circle centred at the origin with radius 13.
13. Write the equation of a circle centred at the origin with radius 14.
14. Write the equation of a circle centred at the origin with radius 15.
15. Write the equation of a circle centred at the origin with radius 0.
16. Write the equation of a circle centred at the origin with radius 9.
17. Write the equation of a circle centred at the origin with radius 17.
18. Write the equation of a circle centred at the origin with radius 18.
19. Write the equation of a circle centred at the origin with radius 19.
20. Write the equation of a circle centred at the origin with radius 20.

Medium

1. What is the radius of the circle with the equation $$x^2 + y^2 = 25$$?
2. What is the radius of the circle with the equation $$x^2 + y^2 = 36$$?
3. What is the radius of the circle with the equation $$x^2 + y^2 = 49$$?
4. What is the radius of the circle with the equation $$x^2 + y^2 = 64$$?
5. What is the radius of the circle with the equation $$x^2 + y^2 = 81$$?
6. What is the radius of the circle with the equation $$x^2 + y^2 = 100$$?
7. What is the radius of the circle with the equation $$x^2 + y^2 = 121$$?
8. What is the radius of the circle with the equation $$x^2 + y^2 = 144$$?
9. What is the radius of the circle with the equation $$x^2 + y^2 = 169$$?
10. What is the radius of the circle with the equation $$x^2 + y^2 = 196$$?
11. What is the radius of the circle with the equation $$x^2 + y^2 = 225$$?
12. What is the radius of the circle with the equation $$x^2 + y^2 = 256$$?
13. What is the radius of the circle with the equation $$x^2 + y^2 = 289$$?
14. What is the radius of the circle with the equation $$x^2 + y^2 = 324$$?
15. What is the radius of the circle with the equation $$x^2 + y^2 = 361$$?
16. What is the radius of the circle with the equation $$x^2 + y^2 = 400$$?
17. What is the radius of the circle with the equation $$x^2 + y^2 = 441$$?
18. What is the radius of the circle with the equation $$x^2 + y^2 = 484$$?
19. What is the radius of the circle with the equation $$x^2 + y^2 = 529$$?
20. What is the radius of the circle with the equation $$x^2 + y^2 = 576$$?

Hard

1. If a circle has the equation $$x^2 + y^2 = 9$$, what is the distance from the origin to any point on the circle?
2. If a circle has the equation $$x^2 + y^2 = 16$$, what is the distance from the origin to any point on the circle?
3. If a circle has the equation $$x^2 + y^2 = 25$$, what is the distance from the origin to any point on the circle?
4. If a circle has the equation $$x^2 + y^2 = 36$$, what is the distance from the origin to any point on the circle?
5. If a circle has the equation $$x^2 + y^2 = 49$$, what is the distance from the origin to any point on the circle?
6. If a circle has the equation $$x^2 + y^2 = 64$$, what is the distance from the origin to any point on the circle?
7. If a circle has the equation $$x^2 + y^2 = 81$$, what is the distance from the origin to any point on the circle?
8. If a circle has the equation $$x^2 + y^2 = 100$$, what is the distance from the origin to any point on the circle?
9. If a circle has the equation $$x^2 + y^2 = 121$$, what is the distance from the origin to any point on the circle?
10. If a circle has the equation $$x^2 + y^2 = 144$$, what is the distance from the origin to any point on the circle?
11. If a circle has the equation $$x^2 + y^2 = 169$$, what is the distance from the origin to any point on the circle?
12. If a circle has the equation $$x^2 + y^2 = 196$$, what is the distance from the origin to any point on the circle?
13. If a circle has the equation $$x^2 + y^2 = 225$$, what is the distance from the origin to any point on the circle?
14. If a circle has the equation $$x^2 + y^2 = 256$$, what is the distance from the origin to any point on the circle?
15. If a circle has the equation $$x^2 + y^2 = 289$$, what is the distance from the origin to any point on the circle?
16. If a circle has the equation $$x^2 + y^2 = 324$$, what is the distance from the origin to any point on the circle?
17. If a circle has the equation $$x^2 + y^2 = 361$$, what is the distance from the origin to any point on the circle?
18. If a circle has the equation $$x^2 + y^2 = 400$$, what is the distance from the origin to any point on the circle?
19. If a circle has the equation $$x^2 + y^2 = 441$$, what is the distance from the origin to any point on the circle?
20. If a circle has the equation $$x^2 + y^2 = 484$$, what is the distance from the origin to any point on the circle?

Answers

Easy

1. $$x^2 + y^2 = 16$$
2. $$x^2 + y^2 = 36$$
3. $$x^2 + y^2 = 25$$
4. $$x^2 + y^2 = 4$$
5. $$x^2 + y^2 = 49$$
6. $$x^2 + y^2 = 64$$
7. $$x^2 + y^2 = 1$$
8. $$x^2 + y^2 = 9$$
9. $$x^2 + y^2 = 100$$
10. $$x^2 + y^2 = 121$$
11. $$x^2 + y^2 = 144$$
12. $$x^2 + y^2 = 169$$
13. $$x^2 + y^2 = 196$$
14. $$x^2 + y^2 = 225$$
15. $$x^2 + y^2 = 0$$
16. $$x^2 + y^2 = 81$$
17. $$x^2 + y^2 = 289$$
18. $$x^2 + y^2 = 324$$
19. $$x^2 + y^2 = 361$$
20. $$x^2 + y^2 = 400$$

Medium

1. 5 units
2. 6 units
3. 7 units
4. 8 units
5. 9 units
6. 10 units
7. 11 units
8. 12 units
9. 13 units
10. 14 units
11. 15 units
12. 16 units
13. 17 units
14. 18 units
15. 19 units
16. 20 units
17. 21 units
18. 22 units
19. 23 units
20. 24 units

Hard

1. 3 units
2. 4 units
3. 5 units
4. 6 units
5. 7 units
6. 8 units
7. 9 units
8. 10 units
9. 11 units
10. 12 units
11. 13 units
12. 14 units
13. 15 units
14. 16 units
15. 17 units
16. 18 units
17. 19 units
18. 20 units
19. 21 units
20. 22 units