Analytic Geometry and Simultaneous Equations
Coordinate geometry
The Basics:
Find the distance between two points using Pythagoras' theorem.
The midpoint is the average (mean) of the coordinates.
The gradient =
Parallel lines have the same gradient. The gradients of perpendicular lines have a product of -1.
Straight Lines:
Equation of a straight line is y = mx + c, where m = gradient, c = y-intercept.
The equation of a line, if we know one point and the gradient is found using:
(y - y1) = m(x - x1)
(If given two points, find the gradient first, and then use the formula.)
Two lines meet at the solution to their simultaneous equations.
Note: When a line meets a curve there will be 0, 1, or two solutions. 1. Use substitution to solve the simultaneous equations 2. Rearrange them to form a quadratic equation 3. Solve the quadratic by factorising, or by using the quadratic formula. 4. Find the y-coordinates by substituting these values into the original equations.
Other Graphs (also in Functions):
Sketch the curve by finding: 1. Where the graph crosses the y-axis. 2. Where the graph crosses the x-axis. 3. Where the stationary points are. 4. Whether there are any discontinuities. 5. What happens as
Circles:
Cartesian equation for a circle is (x - a)2 + (y - b)2 = r2 , where (a, b) is the centre of the circle and r is the radius.
Parametric Equations:
Sketch the graph by substituting in values and plotting points.
Find the cartesian form by either using substitution (use t = ...), or by using the identity, .
Find the gradient using the chain rule:
The Basics:
Find the distance between two points using Pythagoras' theorem.
The midpoint is the average (mean) of the coordinates.
The gradient =
Parallel lines have the same gradient. The gradients of perpendicular lines have a product of -1.
Straight Lines:
Equation of a straight line is y = mx + c, where m = gradient, c = y-intercept.
The equation of a line, if we know one point and the gradient is found using:
(y - y1) = m(x - x1)
(If given two points, find the gradient first, and then use the formula.)
Two lines meet at the solution to their simultaneous equations.
Note: When a line meets a curve there will be 0, 1, or two solutions. 1. Use substitution to solve the simultaneous equations 2. Rearrange them to form a quadratic equation 3. Solve the quadratic by factorising, or by using the quadratic formula. 4. Find the y-coordinates by substituting these values into the original equations.
Other Graphs (also in Functions):
Sketch the curve by finding: 1. Where the graph crosses the y-axis. 2. Where the graph crosses the x-axis. 3. Where the stationary points are. 4. Whether there are any discontinuities. 5. What happens as
Circles:
Cartesian equation for a circle is (x - a)2 + (y - b)2 = r2 , where (a, b) is the centre of the circle and r is the radius.
Parametric Equations:
Sketch the graph by substituting in values and plotting points.
Find the cartesian form by either using substitution (use t = ...), or by using the identity, .
Find the gradient using the chain rule: