# Co-ordinate Geometry Formulas

Co-ordinate Geometry Formulas

co-ordinate geometry is one of the most important part of mathematics.

In co-ordinate geometry we study about the geometry of different objects which are represented as shape made by points in a co-ordinate plane or graph.

Use of co-ordinates in geometry makes geometry more easier to calculate. Some of the important formulas used to formulate equations in co-ordinate geometry are;
1.    In first quadrant, (+, +)
2.    In second quadrant, (-, +)
3.    In third quadrant, (-, -)
4.    In forth quadrant, (+, -)
Where first sign is for x-co-ordinate and second sign for y-co-ordinate

5.    Distance between the line Ax + By + c = 0 and the given ordinate
$\left(x_1, y_1 \right)$ is: $d = \dfrac{Ax_1 + By_1 + C}{\sqrt{A^2 + B^2}}$

6.    When equation Ax + By + C = 0 is reduced in the form of $x\cos \alpha +y \sin \alpha = p$ then the required equation is:
$\dfrac{A}{\pm \sqrt{A^2 + B^2}}x + \dfrac{B}{\pm \sqrt{A^2 + B^2}}y = -\dfrac{C}{\pm \sqrt{A^2 + B^2}}$

7.    The distance between any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$ is given by $AB = d = \sqrt{ \left( x_2- x_1 \right) ^2 + \left( y_2- y_1 \right) ^2}$

8.    The distance between any two points $(x_1, y_1)$ and origin (0, 0) is given by$(d) = \sqrt{(x^2 + y^2)}$

9.    The co-ordinate of the point which divides a straight line joining the given points internally in the ratio $m_1 : m_2$ is given by
$\therefore X = \dfrac{m_1x_2- m_2x_1}{m_2- m_1}, \, Y = \dfrac{m_1y_2- m_2y_1}{m_1- m_2}$ Or,
$\therefore (X, Y) = \left(\dfrac{m_1x_2- m_2x_1}{m_2- m_1}, \dfrac{m_1y_1- m_2y_1}{m_1- m_2}\right)$ is section formula.

10.    The co-ordinate of the point which bisects the line joining two given points $A(x_1, y_1)$ and$B(x_2, y_2)$ is given by
$X = \dfrac{x_1 + x_2}{2}$ and, $Y = \dfrac{y_1 + y_2}{2}$
If the point divides AB in the ratio k:1 then
$X =\dfrac{kx_2 + x_1}{k + 1}$ and, $Y = \dfrac{ky_2 + y_1}{k + 1}$

11.    The co-ordinate of the centroid of a triangle ABC whose vertices are $A(x_1, y_1), B(x_2, y_2)$ and $C(x_3, y_3)$ is the given by
$X = \dfrac{x_1 + x_2 + x_3}{3}$ and $Y = \dfrac{y_1 + y_2 + y_3}{3}$

12.    a. the area $\Delta$ ABC whose vertices are
$A(x_1, y_1), B(x_2, y_2)$ and $C(x_3, y_3)$ is given by

$\Delta ABC = \dfrac{1}{2} \begin{pmatrix}x_1x_2 & x_3x_1 \\ y_1y_3 & y_3y_1\end{pmatrix}$

= $\dfrac{1}{2}[(x_1y_2- x_2y_1) + (x_2y_3- x_3y_2) + (x_3y_1- x_1y_3)]$

b. if one of the vertices of $\Delta ABC$ is the origin
$\Delta ABC = (x_1y_2- x_2y_1)$

13.    let $A (x_1, y_1),B (x_2, y_2),C (x_3, y_3)$ and $D(x_4, y_4)$ be the vertices of a quadrilateral. Then the area of quadrilateral ABCD is

$A = \dfrac{1}{2}[(x_1y_2- x_2y_1) +( x_2y_3- x_3y_2) + (x_3y_4- x_4y_3) + (x_4y_1- x_1y_4)]$

EQUATION OF STRAIGHT LINES

1.    the slope of the line joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by,

a.    slope of the line $m = (\tan\theta) = \dfrac{y_2- y_1}{x_2- x_1}$

b.    when the equation of line is given, (ax + by – c = 0) then slope of line = $\dfrac{-a}{b} = -\dfrac{coefficient of x}{coefficient of y}$

2.    equation of straight line in slope & intercept form is y = mx + c where m is slope and c is x-intercept
3.    when the line passes through the origin, then c = 0 & equation is y = mx.
4.    When the line is parallel to x-axis, then the equation is y = h.
5.    When the line is parallel to y-axis then the equation is x = a.
6.    Equation of straight line in double intercept form is, $\dfrac{x}{a} + \dfrac{y}{b} = 1$
7.    Equation of straight line in normal (perpendicular form is, $x\cos\alpha + y\sin\alpha = P$.

8.    Equation of a line passing from two points $(x_1, y_1)$ and $(x_2, y_2)$ is $(y- y_1) = \dfrac{y_2- y_1}{x_2- x_1}(x- x_1)$

9.    Equation of line passing from a point $(x_1, y_1)$ & slope ‘m’ is, $y- y_1 = m(x- x_1)$

10.    The angle between two lines $y = m_1x + c_1$ and $y = m_2x + c_2$ is $\theta = \tan^{-1}(\dfrac{m_1- m_2}{1 + m_1m_2})$ where $m_1$ = slope of first line & $m_2$ = slope of second line.

11.    When two lines are parallel, then $m_1 = m_2$ (their slope is equal)
12.    When tow lines are perpendicular, then $m_1.m_2 = -1$.

13.    The angle between two lines $A_1x + B_1y + c_1 = 0 \, \& \, Ax_2 + By_2 + c_2 = 0$ is, $\theta = \tan^{-1}(\pm\dfrac{A_2B_1- A_1B_2}{A_1A_2 + B_1B_2})$

PAIR OF LINES

1.    The general equation of any line through the intersection of two lines $A_1x + B_1y + c_1 = 0$ and $A_2x + b_2y + c_2 = 0$ is $(A_1x + B_1y + c_1) + k( A_2x + b_2y + c_2)= 0$ where k is given by $(A_2x_1 + B_2y_1 + C_2) = 0$ and $(x_1, y_1)$ is the point of intersection of the two lines.

2.    If $ax^2 + 2hxy + by^2 = 0$ represents the equation of a pair of straight lines through the origin, then $ax + (h + \sqrt{h^2- ab})y = 0, ax + (h- \sqrt{(h^2- ab)}y = 0$ are the equation of two straight lines represented by the homogeneous equation of the second degree.

3.    The angle between a pair of lines represented by the equation $ax^2 + 2hxy + by^2 = 0$ is given by, $\tan \theta = \pm\dfrac{2\sqrt{h^2- ab}}{a + b}$ if $h^2 = ab, \theta = 0 ^\circ$ i.e. the straight lines are coincident. If $a + b = 0, \theta = 90^\circ$ i.e. the straight lines are perpendicular.

4.    The equation of a circle of radius (r), centre at origin is $x^2 + y^2 = r^2$

5.    The equation of a circle of radius (r), center at (h, k) is $(x- h)^2 + (y- k)^2 = r^2$

6.    Let AB be the diameter of a circle & the co-ordinates of A and B be $(x_1, y_1) \& (x_2, y_2)$. Then the diameter form of equation of a circle is $(y- y_1)(y- y_2) + (x- x_1)(x- x_2) = 0$.

7.    Equation of tangent to the circle $x^2 + y^2 = r^2$ at $(x_1, y_1)$ is $xx_1 + yy_1 = r^2$

8.    Equation of tangent to the circle $x^2 + y^2 + 2hx + 2ky + c = 0$ at $(x_1, y_1)$ is $xx_1 + yy_1 + h(x + x_1) + k (y + y_1) +c = 0$
The second degree general equation which may represent pair of straight lines
= $ax^2 + 2hxy + by^2 + 2gx + 2fy +c = 0$
=$ax^2 + 2hxy + by^2 = 0$

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