## Cartesian Equation of a Circle

Since a circle is made up of all the points a fixed distance (its radius) from a given point (its centre) then the equation of a circle simply needs to ensure this is true. This can be done using Pythagoras’s theorem. This is because we can draw a right angled triangle with the centre of the circle at one corner and the point on the circle at the opposite corner as shown below. The radius is then the hypotenuse, the vertical side is the difference between the y co-ordinate of the point and that of the centre and the horizontal side is the difference between the x co-ordinate of the point and centre. From Pythagoras we therefore know that a circle of radius r and centre (a,b) must have a Cartesian equation

**r ^{2} = (x-a)^{2} + (y-b)^{2}**

However, we can expand these brackets out to get

r^{2} = x^{2} – 2ax + a^{2} + y^{2} – 2by + b^{2}

but since a^{2}+b^{2}+r^{2}, -a and -b are all constant we can let

c = a^{2}+b^{2}-r^{2},

g = -a

f = -b

to get

**x ^{2} + y^{2} + 2gx + 2fy + c = 0**

where the circle has a centre (-g,-f) and radius √(a^{2}+b^{2}-c^{2})

## Equation of a Line: y = mx + c

The equation of all straightlines can be written in the form

**y=mx+c**

where m is the gradient, c is the intercept on the y-axis and you are plotting xagainst y. m and c ae constants meaning their value is fixed.

The y intercept, c, is how far up, or down if its negative, he line crosses the veritcal yaxis.

The gradient, m, is how step the line is. A gradient equal to 1 means that the graph is at 45 degress to the axis, a gradient greater than 1 is steeper and a gradient less than 1 is shallower.

To find where the graph cuts the x-axis simply let y=0 and find the value of x since at the x-axis the grap has zero height so y=0.

The easiest way to sketch a graph of the form y=mx + c is to find the x and y intercepts and then draw a straight line through these points.

### Find the gradient of a line

The gradient, m, of a straight line can be found using the equation

m=change in y/change in x = (y_{1}-y_{2})/(x_{1}-x_{2})

Where x1 and y1 are the co-ordinates of one point on the line and x2 and y2 are the coordinate of another point on the line.

If you have any comments, questions or improvements please them as a comment below.

**By David Woodford**

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