Two real number lines that are perpendicular to each other and that intersect at their respective zero points define a **rectangular coordinate system**. Often called the **x-y coordinate system** or **x-y plane**. The horizontal number line is called the **x-axis** and the vertical number line is called the **y-axis**. The point where the two axes intersect is called the **origin**, denoted by O. The positive half of the x-axis is to the right of the origin, and the positive half of the y-axis is above the origin. The two axes divide the plane into four regions called **quadrants I**, **II**, **III**, and **IV**, as shown in the figure below.

Each point P in the x-y plane can be identified with an ordered pair (x, y) of real numbers and is denoted by P (x, y). The first number is called the x-coordinate, and the second number is called the y-coordinate. A point with coordinates **(x, y)** is located **lxl** units to the right of the y-axis if x is positive or to the left of the y-axis if x is negative. Also, the point is located **lyl** units above the x-axis if y is positive or below the x-axis if y is negative. If x = 0, the point lies on the y-axis, and if y = 0, the point lies on the x-axis. The origin has coordinates (0, 0).

In the figure above, the point P (4, 2) is 4 units to the right of the y-axis and 2 units above the x-axis, and the point P’’’ (-4, -2) is 4 units to the left of the y-axis and 2 units below the x-axis.

**Distance Formula:**

If Two points P and Q are such that they are represented by the points (x_{1}, y_{1}) and (x_{2}, y_{2}) on the x-y plane, then the distance between two pints P and Q = **((x _{1 }– x_{2})^{2} + (y_{1 }– y_{2})^{2})^{0.5}**

**Section Formula:**

** **If any point C (x, y) divides the line segment joining the points A (x_{1}, y_{1}) and B (x_{2, }y_{2}) in the ratio m: n internally,

X = (mx_{2} + nx_{1})/ (m + n)

Y = (my_{1 }+ my_{2})/ (m + n)

If any point C (x, y) divides the line segment joining the points A (x_{1}, y_{1}) and B (x_{2, }y_{2}) in the ratio m: n externally,

X = (mx_{2} – nx_{1})/ (m + n)

Y = (my_{1 }– my_{2})/ (m + n)

**Slope of a line:**

** **The slope of a line joining two points A (x_{1}, y_{1}) and B (x_{2}, y_{2}) is denoted by m and is given by m = (y_{2 }– y_{1})/ (x_{2 }– x_{1}) = tan , where is the angle that the line makes with the positive direction of x-axis.

**Equation of line:**

**Slope Intercept form:**

The equation of a straight line passing through the point A (x_{1}, y_{1}) and having slope m is given by

(y – y_{1}) = m (x – x_{1})

**Intercept form:**

The equation of a straight line making intercepts a and b on the axes of x and y respectively is given by

**x/a + y/b = 1**

** **

**Perpendicularity and parallelism:**

**Conditions for two lines to be parallel:**

Two lines are said to be parallel if their slopes are equal.

**Conditions for two lines to be perpendicular:**

Two lines are said to be perpendicular if the product of slopes of two lines is equal to -1.