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

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

## Straight linesEdit

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 lineEdit

The most common form of a straight line is $y=mx + c$. 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 $x = a$, used for vertical lines of infinte gradient, $y = b$, used for horizontal lines with 0 gradient, and $px + qy + r = 0$, which is often used for some lines as a neater way of writing the equation.

#### Finding the equation of a straight lineEdit

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 $({x_1}, {y_1})$, and the co-ordinates and the gradient can be substituted in the formula $y-{y_1} = m(x-{x_1})$. Then it is simply a case of rearranging the formula into the form $y=mx + c$.

You may only be given two points, $({x_1}, {y_1})$ and $({x_2}, {y_2})$. In this case, use the formula $m = \frac {{y_2} - {y_1}} {{x_2} - {x_1}}$ 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.

$m=\frac {y_2-y_1}{x_2-x_1}$

#### Parallel and perpendicular linesEdit

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, $m_1=m_2$. A pair of lines are perpendicular if the product of their gradients is -1, $m_1 \times m_2=-1$

### Distance between two pointsEdit

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$({x_1},{y_1})$ and B$({x_2},{y_2})$ is given by $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

### Mid-point of a lineEdit

When the co-ordinate of two points are known, the mid-point is the point halfway between those points. For any two points A$({x_1},{y_1})$ and B$({x_2},{y_2})$, the co-ordinates of the mid-point of AB can be found by $\left(\frac {{x_1} + {x_2}}{2}, \frac {{y_1} + {y_2}}{2}\right)$.

### Intersection of linesEdit

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.

## CurvesEdit

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 $y=x^n$Edit

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

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

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

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

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

(Images here)

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

### Intersection of lines and curvesEdit

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 $y = mx + c$, then you can replace any instances of $y$ with $mx + c$, and then expand the equation out and then factorise the resulting quadratic.

### Intersection of curvesEdit

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 circleEdit

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 circleEdit

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

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

### Circle geometryEdit

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