# A-level Mathematics/MEI/C1/Co-ordinate Geometry

< A-level Mathematics‎ | MEI‎ | C1

Co-ordinates are a way of describing position. In two dimensions, positions are given in two perpendicular directions, x and y.

## Straight lines

A straight line has a fixed gradient. The gradient of a line and its y intercept are the two main pieces of information that distinguish one line from another.

### Equations of a straight line

The most common form of a straight line is ${\displaystyle y=mx+c}$. The m is the gradient of the line, and the c is where the line intercepts the y-axis. When c is 0, the line passes through the origin.

Other forms of the equation are ${\displaystyle x=a}$, used for vertical lines of infinte gradient, ${\displaystyle y=b}$, used for horizontal lines with 0 gradient, and ${\displaystyle px+qy+r=0}$, which is often used for some lines as a neater way of writing the equation.

#### Finding the equation of a straight line

You may need to find the equation of a straight line, and only given the co-ordinates of one point on the line and the gradient of the line. The single point can be taken as ${\displaystyle ({x_{1}},{y_{1}})}$, and the co-ordinates and the gradient can be substituted in the formula ${\displaystyle y-{y_{1}}=m(x-{x_{1}})}$. Then it is simply a case of rearranging the formula into the form ${\displaystyle y=mx+c}$.

You may only be given two points, ${\displaystyle ({x_{1}},{y_{1}})}$ and ${\displaystyle ({x_{2}},{y_{2}})}$. In this case, use the formula ${\displaystyle m={\frac {{y_{2}}-{y_{1}}}{{x_{2}}-{x_{1}}}}}$ to find the gradient and then use the method above.

The steepness of a line can be measured by its gradient, which is the increase in the y direction divided by the increase in the x direction. The letter m is used to denote the gradient.

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

#### Parallel and perpendicular lines

With the gradients of two lines, you can tell if they are parallel, perpendicular, or neither. A pair of lines are parallel if their gradients are equal, ${\displaystyle m_{1}=m_{2}}$. A pair of lines are perpendicular if the product of their gradients is -1, ${\displaystyle m_{1}\times m_{2}=-1}$

### Distance between two points

Using the co-ordinates of two points, it is possible to calculate the distance between them using Pythagoras' theorem.

The distance between any two points A${\displaystyle ({x_{1}},{y_{1}})}$ and B${\displaystyle ({x_{2}},{y_{2}})}$ is given by ${\displaystyle {\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}$

### Mid-point of a line

When the co-ordinate of two points are known, the mid-point is the point halfway between those points. For any two points A${\displaystyle ({x_{1}},{y_{1}})}$ and B${\displaystyle ({x_{2}},{y_{2}})}$, the co-ordinates of the mid-point of AB can be found by ${\displaystyle \left({\frac {{x_{1}}+{x_{2}}}{2}},{\frac {{y_{1}}+{y_{2}}}{2}}\right)}$.

### Intersection of lines

Any two lines will meet at a point, as long as they are not parallel. You can find the point of intersection simply by solving the two equations simultaneously. The lines will intersect at one distinct point (if a solution to their equation exists) or will not intersect at all (if they are parallel). A curve can however intersect a line or another curve at multiple points.

## Curves

To sketch a graph of a curve, all you need to know is the general shape of the curve and other important pieces of information such as the x and y intercepts and the points of any maxima and minima.

### Curves in the form ${\displaystyle y=x^{n}}$

Here are the graphs for the curves ${\displaystyle y=x^{1}}$, ${\displaystyle y=x^{2}}$, ${\displaystyle y=x^{3}}$ and ${\displaystyle y=x^{4}}$:

(Need to draw those later, just simple b&w curve sketches for each curve)

Notice how the odd powers of ${\displaystyle x}$ all share the same general shape, moving from bottom-left to top-right, and how all the even powers of ${\displaystyle x}$ share the same "bucket" shaped curve.

### Curves in the form ${\displaystyle y={\frac {1}{x^{n}}}}$

Just like earlier, curves with an even powers of ${\displaystyle x}$ all have the same general shape, and those with odd powers of ${\displaystyle x}$ share another general shape.

(Images here)

All curves in this form do not have a value for ${\displaystyle x=0}$, because ${\displaystyle {\frac {1}{0}}}$ is undefined. There are asymptotes on both the ${\displaystyle x}$ and ${\displaystyle y}$ axis, where the curve moves towards increasingly slowly but will never actually touch.

### Intersection of lines and curves

When a line intersects with a curve, it is possible to find the points of intersection by substituting the equation of the line into the equation of the curve. If the line is in the form ${\displaystyle y=mx+c}$, then you can replace any instances of ${\displaystyle y}$ with ${\displaystyle mx+c}$, and then expand the equation out and then factorise the resulting quadratic.

### Intersection of curves

The same method can be used as for a line and a curve. However, it will only work in simple cases. When an algebraic method fails, you will need to resort to a graphical or Numerical Method. In the exam, you will only be required to use algebraic methods.

## The circle

The circle is defined as the path of all the points at a fixed distance from a single point. The single point is the centre of the circle and the fixed distance is it's radius. This definition is the basis of the equation of the circle.

### Equation of the circle

The equation of the circle is ${\displaystyle {x^{2}}+{y^{2}}=r^{2}}$ for a circle center (0,0) and radius r, and ${\displaystyle {(x-a)^{2}}+{(y-b)^{2}}=r^{2}}$ for a circle centre (a,b) and radius r.

So, for example, a circle with the equation ${\displaystyle {(x+2)^{2}}+{(y-3)^{2}}=25}$ would have centre (-2,3) and radius 5.

### Circle geometry

When presented with a problem, it may appear at first that there is not enough information given to you. However, there are some facts that will help you spot right angles in relation to a circle.

• The angle in a semi-circle is a right angle
• The perpendicular from the centre of a circle to a chord bisects the chord
• The tangent to a circle at a point is perpendicular to the radius through that point