Coordinate Geometry
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Coordinate geometry is a branch of mathematics in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic. It springs from the idea that any point in two-dimensional space can be represented by two numbers and any point in three-dimensional space by three. Because lines, circles, spheres, and other figures can be thought of as collections of points in space that satisfy certain equations, they can be explored via equations and formulas rather than graphs. Its most common application is in the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions. For example, the linear equation ax+by+c=0 represents a straight line in the xy-plane, and the linear equation ax+by+cz+d=0 represents a plane in space, where a, b, c, and d are constant numbers (coefficients). In this way a geometric problem can be translated into an algebraic problem and the methods of algebra brought to bear on its solution. Conversely, the solution of a problem in algebra, such as finding the roots of an equation or system of equations, can be estimated or sometimes given exactly by geometric means, e.g., plotting curves and surfaces and determining points of intersection.
COORDINATE GEOMETRY
SOME BASIC FORMULAE
1. DISTANCE BETWEEN TWO POINTS
Let there be two points A(x1, y1) and B(x2,y2). Then the distance between them is given by
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AB = {(x2-x1)2 + (y2-y1)2}1/2 |
2. SECTION FORMULA
To find the co-ordinates of a point P which divides the line segment AB in the ratio m1:m2 where co-ordinates of A are (x1,y1) and that of B are (x2,y2)
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x = (m1x2+m2x1)/(m1+m2) y = (m1y2+ m2y1)/(m1+m2) |
This formula is valid only for an internal division of the line. In the case of an external division we use the formula
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x = (m1x2-m2x1)/(m1-m2) y = (m1y2 - m2y1)/(m1-m2) |
MID-POINT
Putting m1 m2 in the above equations we find that the mid-point of the line joining the points (x1, y1), (x2,y2) is
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x= (x1+x2)/2 y= (y1+y2)/2 |
It has been assumed that m1 and m2 are positive numbers, if however, we take AP:PB to be positive in the former case and negative in the latter case, we have in both cases the formula
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x = (m1x2+m2 x1) / (m1+m2 ) y = (m1y2+m2 y1) / (m1+m2 ) |
AREA OF A TRIANGLE
To find the area of a triangle formed by three points A(x1,y1), B(x2,y2) and C(x3,y3)
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Area = 1/2 (x1y2- x2y1 + x2y3 - x3y2+ x3y1 - x1y3) |
NOTE : CONDITION FOR CO-LINEARITY
If the Area of a triangle formed by the three points is zero then it implies that the points are collinear.