Coordinate Geometry

Equations of Straight Lines

The general equation of a straight line is given by:

$y = mx + c$

where m is the gradient and c is the y-intercept.

Alternatively, the equation can be written in the form:

$ax + by + d = 0$

where a, b, and d are constants.

The gradient m can be calculated using the two-point form:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are two distinct points on the line.

Different Forms of the Equation

• Slope-intercept form: $y = mx + c$
• Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$ (where a is the x-intercept and b is the y-intercept)
• Point-slope form: $y - y_1 = m(x - x_1)$ (where $(x_1, y_1)$ is a point on the line)
• Standard form: $ax + by + d = 0$

Intersections

Intersection of Two Straight Lines

To find the point of intersection of two straight lines, we need to solve the system of equations:

1. $y = m_1x + c_1$
2. $y = m_2x + c_2$

Setting the two expressions for y equal to each other gives:

$m_1x + c_1 = m_2x + c_2$

$m_1x - m_2x = c_2 - c_1$

$x(m_1 - m_2) = c_2 - c_1$

$x = \frac{c_2 - c_1}{m_1 - m_2}$

Substitute the value of x into either of the original equations to find the corresponding value of y.

Intersection of a Line and a Circle

To find the points of intersection between a straight line and a circle, substitute the equation of the line into the equation of the circle. This will result in a quadratic equation in terms of x. Solve the quadratic equation to find the values of x, and then substitute these values back into the equation of the line to find the corresponding values of y.

Midpoints

The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Gradient

The gradient of a straight line represents its steepness and direction. It is calculated as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line.

A positive gradient indicates a line that slopes upwards from left to right.

A negative gradient indicates a line that slopes downwards from left to right.

A gradient of zero indicates a horizontal line.

An undefined gradient indicates a vertical line.

Circles

The equation of a circle with centre $(a, b)$ and radius r is given by:

$(x - a)^2 + (y - b)^2 = r^2$

The centre of the circle is $(a, b)$ and the radius is r.

The equation of a circle can also be written in general form:

$x^2 + y^2 + 2ax + 2by + c = 0$

where the centre is $(-a, -b)$ and the radius is $\sqrt{a^2 + b^2 - c}$.

Intersection of a Line and a Circle

As mentioned earlier, to find the intersection points of a line and a circle, substitute the equation of the line into the equation of the circle and solve the resulting quadratic equation.

Topic	Formula	Description
Gradient	$m = \frac{y_2 - y_1}{x_2 - x_1}$	Change in y divided by change in x between two points.
Midpoint	$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$	The point halfway between two given points.
Equation of a Line (Slope-intercept)	$y = mx + c$	The standard form of a linear equation.
Equation of a Circle	$(x - a)^2 + (y - b)^2 = r^2$	The standard form of a circle equation.