Introduction to Coordinate Geometry
Coordinate geometry is a branch of mathematics that uses a coordinate system to describe the position of a point in a plane.
- This system was largely inspired by the work of the French mathematician René Descartes in the \(17\)th century.
- The idea is to use two perpendicular number lines to locate any point in a plane.
The Cartesian System and Co-ordinate Axes:
The system used to describe the position of a point in two dimensions is called the Cartesian Co-ordinate System or the \(xy\)-plane.
Co-ordinate Axes:
- The horizontal number line is called the \(x\)-axis (represented by \(X'OX\)).
- The vertical number line is called the \(y\)-axis (represented by \(Y'OY\)).
Origin:
The point where the \(x\)-axis and \(y\)-axis intersect is called the Origin, represented by the coordinates \((0, 0)\).
Sign Convention:
- On the \(x\)-axis, points are positive along \(OX\) (right) and negative along \(OX'\) (left).
- On the \(y\)-axis, points are positive along \(OY\) (up) and negative along \(OY'\) (down).
Co-ordinates and Ordered Pairs:
The position of a point is denoted by an ordered pair $(a, b)$. The order matters; \((a, b)\) is not the same as \((b, a)\).
\(x\)-coordinate (\(a\)):
- The distance of the point along the \(x\)-axis from the origin.
- It is also known as the abscissa.
\(y\)-coordinate (\(b\)):
- The distance of the point along the \(y\)-axis from the origin.
- It is also known as the ordinate.
Example:
To plot the point \((2, 5)\), you move \(2\) units along the positive \(x\)-axis and then \(5\) units parallel to the positive \(y\)-axis.
Quadrants:
The \(x\)-axis and \(y\)-axis divide the Cartesian plane into four regions called quadrants, which are numbered in an anticlockwise direction starting from the region bounded by the positive \(x\) and \(y\) axes (\(XOY\)).
- Quadrant I: Any point located in quadrant I will have a positive number in the \(x\) - axis and \(y\) - axis. i.e, \(x>0, y>0\)
- Quadrant II:Any point located in quadrant II will have a negative number in the \(x\) - axis and a positive number in \(y\) - axis. i.e., \(x<0, y>0\)
- Quadrant III:Any point located in quadrant III will have a negative number in the \(x\) - axis and \(y\) - axis. i.e., \(x<0, y<0\).
- Quadrant IV:Any point located in quadrant IV will have a positive number in the \(x\) - axis and a negative number in \(y\) - axis. i.e., \(x>0, y<0\).
Points on the Axes:
A point is located on an axis if one of its coordinates is zero.
- Point lies on the \(x\)-axis: The \(y\)-coordinate is zero. The point is represented as \((x, 0)\).
- Point lies on the \(y\)-axis: The \(x\)-coordinate is zero. The point is represented as \((0, y)\).
Lines Parallel to the Axes:
Line parallel to the \(x\)-axis:
- The distance from the \(x\)-axis is constant.
- The line is represented by the equation \(y = c\), where \(c\) is a constant.
Line parallel to the \(y\)-axis:
- The distance from the \(y\)-axis is constant.
- The line is represented by the equation \(x = c\), where \(c\) is a constant.