0.2: Analytic geometry
This page is a draft and is under active development.
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The straight line
Theorem 0.2.1
Given two points A=(x1,y1) and B=(x2,y2),
1. The slope of AB=y2−y1x2−x1= rise run.
2. The length of the line segment AB=√(x2−x1)2+(y2−y1)2.
3. The mid-point C, of the line AB has coordinates ((x1+x2)2,(y1+y2)2).
It is often useful to be able to describe the path or locus of a point as a relationship between the coordinates of an arbitary point on the path. Some of the standard paths are described below. It is worth noting here that the equation of a straight line is completely determined if we know the slope of the line and one point on the line.
Equations of the straight line
Equations of the straight line
1. The point slope form
The equation of the straight line which has slope m and which passes through the point (x1,y1) is, y−y1=m(x−x1).
2. The standard form
The equation of the straight line which has slope m and makes an intercept of b units on the y−axis is y=mx+b.
Any straight line in the plane has an equation of the form ax+by+c=0.
Where a,b and c are real numbers. This straight line has slope −ab and y−intercept −cb, provided b≠0.
Parallel straight lines and perpendicular straight lines
Slopes
1. Two straight lines are parallel if and only if their slopes are equal.
If L1 is a straight line with slope m1 and L2 is a straight line with slope m2, and if neither line is parallel to either x−axis or the y−axis, then L1 is perpendicular to L2 if and only if m1m2=−1.
(Alternatively we note that the slope of L1 is the negative reciprocal of slope of L2.)
2. The perpendicular distance of a point (h,k) from the straight line with equation ax+by+c=0 is given by
d=|ah+bk+c|√a2+b2.
Though it is not essential to memorize this formula, it can sometimes be very useful.
The circle
Circle
1. The equation of a circle with the centre (h,k) and the radius r is (x−h)2+(y−k)2=r2.
2. The general form of the equation of a circle is x2+y2+2gx+2fy+c=0,
The conic sections
Conic Sections
The general form of the equation of the conic section which has axes parrallel to the x− and y− axes is
ax2+by2+cx+dy+e=0,
provided that a and b are not both zero.
The conic sections may be readily identified using the information given below:
Equation ~??? represents
1. a circle provided that a=b.
2. a parabola with axis parallel to the x−axis if a=0, and b≠0.
3. a parabola with axis parallel to the y−axis if b=0, and a≠0.
4. an ellipse provided that a≠b and both a and b have the same sign.
5. a hyperbola provided that a and b differ in sign.
6. two straight lines exactly when bc2−ad2−4abe=0.