Circle with Centre (0,0) Revision Notes for Leaving Cert Mathematics

Circle with Centre (0,0)

What is a circle with centre (0,0)?

A circle with centre (0,0) is a circle positioned at the origin of a coordinate system. The origin is the point where the x-axis and y-axis meet, with coordinates (0,0).

In this diagram, we can see a circle centred at the origin with a point P(x,y) marked on its circumference. The distance from the centre to any point on the circle is called the radius (r).

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The coordinate system uses the origin (0,0) as the central reference point where both axes intersect. This makes calculations simpler when the circle is positioned at this special location.

Deriving the equation

To understand how we get the equation of a circle with centre (0,0), we need to think about what makes a circle. A circle is the set of all points that are the same distance (radius) from its centre.

When we have a point P(x,y) on the circle, we can form a right-angled triangle:

The horizontal side has length x
The vertical side has length y
The hypotenuse is the radius r

Using Pythagoras' theorem on this right triangle: $x^2 + y^2 = r^2$

This gives us the fundamental equation for a circle with centre (0,0).

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The beauty of this derivation is that it connects the geometric concept of a circle (constant distance from centre) with the algebraic representation using Pythagoras' theorem.

Key formula

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The equation of a circle with centre (0,0) and radius r is: $x^2 + y^2 = r^2$

This is the standard form that you must remember and be able to apply in all problems.

To find the equation of any circle with centre (0,0), we need to know:

The centre coordinates (which are always (0,0) for this type)
The length of the radius

Finding the equation when given the radius

When you know the radius, simply substitute it into the formula.

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Worked Example 1: Finding equations from given radii

Find the equation of the circle with centre (0,0) and radius: (i) 3 (ii) 1¼

Solution: (i) Using $x^2 + y^2 = r^2$ with r = 3: $x^2 + y^2 = 3^2 = 9$

(ii) Here r = 1¼ = 5/4, so: $x^2 + y^2 = \left(\frac{5}{4}\right)^2 = \frac{25}{16}$

We can multiply both sides by 16 to get: $16x^2 + 16y^2 = 25$

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Worked Example 2: Finding equation from a point on the circle

Find the equation of the circle with centre (0,0) that contains the point (4, -1).

Solution: The radius equals the distance from (0,0) to (4, -1).

Using the distance formula: $r = \sqrt{(4-0)^2 + (-1-0)^2} = \sqrt{16 + 1} = \sqrt{17}$

Therefore, the equation is: $x^2 + y^2 = (\sqrt{17})^2 = 17$

Finding the radius when given the equation

When you have the equation in the form $x^2 + y^2 = r^2$ , the radius is simply $r = \sqrt{r^2}$ .

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Critical Step: If the equation is not in standard form, you must rearrange it first before identifying the radius.

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Worked Example 3: Finding radius from equations

Find the radius of these circles: (i) $x^2 + y^2 = 8$ (ii) $9x^2 + 9y^2 = 16$

Solution: (i) From $x^2 + y^2 = 8$ : $r = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$

(ii) First, divide both sides by 9: $x^2 + y^2 = \frac{16}{9}$

Therefore: $r = \sqrt{\frac{16}{9}} = \frac{4}{3}$

Exam tips

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Essential Exam Strategies:

Always check if the equation is in standard form $x^2 + y^2 = r^2$
If coefficients appear in front of $x^2$ and $y^2$ , divide the entire equation to get them equal to 1
Remember that $(\sqrt{a})^2 = a$ when working with radii
The centre is always (0,0) for this type of circle equation
Use the distance formula to find radius when given a point on the circle

Remember!

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Key Points to Remember:

The equation of a circle with centre (0,0) is $x^2 + y^2 = r^2$
The radius can be found from the equation as $r = \sqrt{r^2}$
When finding the equation from a given point, use the distance formula to calculate the radius first
Always ensure the equation is in standard form before identifying the radius
The origin (0,0) means the circle is centred where the axes meet

Circle with Centre (0,0) (Leaving Cert Mathematics): Revision Notes