Also read, |

Circle Formulas |

Area Formulas |

Lines and Angles |

Straight Line |

Conic Section Formula |

# Coordinate geometry:

To study coordinate geometry formula we must know about quadrants. As there are for quadrants each has its own sign rule.

**Quadrants:**

All the four quadrants have their respective values with respective signs:

- 1
^{st}Quadrant : (+x, +y) - 2
^{nd}Quadrant : (-x, +y) - 3
^{rd}Quadrant : (-x, -y) - 4
^{th}Quadrant : (+x, -y)

**Distance between two points:**

If A ( x_{1}, y_{1}) and B( x_{2}, y_{2},), then distance d, from A to B =

\(\large d = \sqrt{(x_{2} -x_{1})^{2} + (y_{2} – y_{1})^{2}}\)

**Midpoint of a line joining two points:**

If A (x_{1}, y_{1}) and B( x_{2}, y_{2},) represent the end point of line and M lies at midpoint of the line. Thus,

M (x, y) = \(\left ( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right )\)

**Representation of Equation of line:**

We can represent equation of a line in many ways, few of them is here-

**General form-**The general form of a line is given as Ax + By + C = 0.**Slope intercept form-**Let us assume that x, y be the coordinate of a point which a line passes through, m be the slope of a line, and c be the y-intercept, then the equation of a line is given as- y=mx + c**Intercept form of a line-**Let x-intercept and y-intercept of a line be a and b respectively, then the equation of a line is represented as-

\(\large \boldsymbol{\frac{x}{a} + \frac{y}{b} = 1}\)

**Coordinate geometry formula for slope of a line:**

It can be obtained as per provided information as,

**From General Form:
**Ax+By+C=0 ⇒ By=−Ax–C

\(\large \Rightarrow y = -\frac{A}{B}x – \frac{C}{B}\)Let’s compare the above equation with y = mx + c,

we found that

\(\large \boldsymbol{m = -\frac{A}{B}}\)

**Angle between two lines:**

Let two lines be A and B, having their slopes to be m1 & m2 respectively.

Let θ be the angle between these two lines, then the angle between them can be represented as-

tan θ = \(\frac{m_{1} – m_{2}}{1 + m_{1} m_{2}}\)

**Case1. **

When the two lines are parallel to each other then m_{1 }= m_{2} = m

**Case2.
**When the two lines are perpendicular to each other than m

_{1}. m

_{2 }= -1

**Section Formula:**

Section formula is a base part of coordinate geometry formula. In a line with two end points A and B having coordinates (x_{1},y_{1}) and (x_{2}, y_{2}) Also M be any point collinear with the same line. The coordinates of M

**Case 1.
**When the ratio m:n is internally:

\(\large \left (\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n} \right )\)

**Case2.
**When the ratio m:n is externally:

\(\large \left (\frac{mx_{2} – nx_{1}}{m – n}, \frac{my_{2} – ny_{1}}{m – n} \right )\)

**Area of Triangle:**

In coordinate geometry formula we also study about the area of the triangle. Thus the area of triangle whose vertices are (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is:

\(\frac{1}{2}\) | X_{1} (Y_{2} – Y_{3}) + X_{2} (Y_{3} – Y_{1}) + X_{3} (Y_{1} – Y_{2}) |

If the area of a triangle whose vertices are (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is zero, then the three points are collinear.

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