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 $y=mx + c$. 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 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 $({x_1}, {y_1})$, 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 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, $m_1=m_2$. A pair of lines are perpendicular if the product of their gradients is -1, $m_1 \times m_2=-1$

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

Mid-point of a 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$({x_1},{y_1})$ 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 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 $y=x^n$

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 $y=\frac {1} {x^n}$

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 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 $y = mx + c$, 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 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 ${x^2} + {y^2} = r^2$ 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 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