# Analytic Geometry and Simultaneous Equations

**Topics:**Analytic geometry, Euclidean geometry, Cartesian coordinate system

**Pages:**1 (262 words)

**Published:**February 2, 2013

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:

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