Year 11 Maths: Graph circles centered at the origin

Today, we are going to learn about graphing circles centred at the origin. This might seem a bit tricky at first, but don’t worry – we’ll walk through it step by step and you’ll soon get the hang of it!

What is a circle?

Firstly, let’s go back to basics. What is a circle? A circle is a shape where all points are the same distance from a central point – the centre.

Understanding the equation

In maths, we can describe circles using equations. The general equation for a circle centred at the origin (0,0) is:

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

Where r is the radius of the circle. The radius is the distance from the centre of the circle to any point on the circle.

Graphing the circle

To graph a circle, first find the radius by solving the equation for r. Then draw the circle by plotting points that are r units away from the origin in all directions.

Now, let’s try an example:

Let’s say we have the equation $$x^2 + y^2 = 9$$ . To find the radius, we solve for r:

$$r^2 = 9$$

$$r = 3$$

So the radius of our circle is 3 units. Now we can draw our circle by plotting points that are 3 units away from the origin in all directions.

Tips and Tricks

• Remember, the centre of the circle is always at the origin (0,0).
• The radius of the circle is the square root of the number on the other side of the equation.
• To check your work, you can plot a few points and see if they form a circle.

Practice Questions

Easy Level

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

Medium Level

1. Find the radius and graph the circle with the equation $$x^2 + y^2 = 121$$
2. Find the radius and graph the circle with the equation $$x^2 + y^2 = 144$$
3. Find the radius and graph the circle with the equation $$x^2 + y^2 = 169$$
4. Find the radius and graph the circle with the equation $$x^2 + y^2 = 196$$
5. Find the radius and graph the circle with the equation $$x^2 + y^2 = 225$$
6. Find the radius and graph the circle with the equation $$x^2 + y^2 = 256$$
7. Find the radius and graph the circle with the equation $$x^2 + y^2 = 289$$
8. Find the radius and graph the circle with the equation $$x^2 + y^2 = 324$$
9. Find the radius and graph the circle with the equation $$x^2 + y^2 = 361$$
10. Find the radius and graph the circle with the equation $$x^2 + y^2 = 400$$

Hard Level

1. Given the equation $$x^2 + y^2 = 25$$ , find the points where the circle intersects the x-axis.
2. Given the equation $$x^2 + y^2 = 36$$ , find the points where the circle intersects the y-axis.
3. Given the equation $$x^2 + y^2 = 64$$ , find the points where the circle intersects the x-axis.
4. Given the equation $$x^2 + y^2 = 81$$ , find the points where the circle intersects the y-axis.
5. Given the equation $$x^2 + y^2 = 100$$ , find the points where the circle intersects the x-axis.
6. Given the equation $$x^2 + y^2 = 121$$ , find the points where the circle intersects the y-axis.
7. Given the equation $$x^2 + y^2 = 144$$ , find the points where the circle intersects the x-axis.
8. Given the equation $$x^2 + y^2 = 169$$ , find the points where the circle intersects the y-axis.
9. Given the equation $$x^2 + y^2 = 196$$ , find the points where the circle intersects the x-axis.
10. Given the equation $$x^2 + y^2 = 225$$ , find the points where the circle intersects the y-axis.

Answers

Easy Level

The graphs will be circles with radii 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 units respectively.

Medium Level

The radii will be 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20 units respectively. The graphs will be circles with these radii.

Hard Level

1. The points are (-5,0) and (5,0).
2. The points are (0,-6) and (0,6).
3. The points are (-8,0) and (8,0).
4. The points are (0,-9) and (0,9).
5. The points are (-10,0) and (10,0).
6. The points are (0,-11) and (0,11).
7. The points are (-12,0) and (12,0).
8. The points are (0,-13) and (0,13).
9. The points are (-14,0) and (14,0).
10. The points are (0,-15) and (0,15).