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Coordinate Geometry

Notes

Formulas

1. Distance

If A ( x1, y1) and B( x2, y2,), then distance d, from A to B


2. Midpoint



3. Gradient/Slope



4. Equation of a line

y = mx + c

y - y1 = m(x - x1)

5. Parallel lines

If two lines are parallel, then they have the same gradient.

6. Perpendicular lines

If two lines are perpendicular, then the product of the gradients of the two lines is -1.

or: perpendicular gradient = -1/m    where m is the gradient of the line perpendicular to it. 

7. Area of triangle

The area of the triangle formed by the three points (x1, y1), (x2, y2), (x3, y3)


8. Shoelace formula

  • go anti-clockwise direction
  • must go back to first coordinate

9. Circle

The equation of a circle whose center is (h,k) and radius is a is given by the equation

(x - h)2 + (y - k)2 = 0


The equation of a circle whose centre is the origin and whose radius is a is given by the equation


x2 + y2 = a2


The general equation of a circle is

x2 + y2 + 2gx + 2fy + c = 0

where the centre is (-g,-f) and radius is


The equation of a circle whose one diameter is the line segment joining the points
(x1, y1), (x2, y2) is given by

(x - x1)(x - x2) + (y - y1)(y - y2) = 0

Example

1. Find the equation of the line with gradient 2 passing through (1, 4).

y - 4 = 2(x - 1)
y - 4 = 2x - 2
y = 2x + 2

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