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Distance Formula
For two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) in the Cartesian plane, the distance between them is given by:
\[
D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
\]
Section Formula
For two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the coordinates ( (x, y) ) of a point ( P ) dividing the line segment ( AB ) internally in the ratio ( m:n ) are:
\[
x = \frac{m x_2 + n x_1}{m + n}
\]
\[
y = \frac{m y_2 + n y_1}{m + n}
\]
For two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the coordinates ( (x, y) ) of a point ( P ) dividing the line segment ( AB ) externally in the ratio ( m:n ) are:
\[
x = \frac{m x_2 – n x_1}{m – n}
\]
\[
y = \frac{m y_2 – n y_1}{m – n}
\]
Mid Point Formula
For two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the coordinates ( (x, y) ) of a mid point ( P ) is :
\[
x = \frac{x_2 + x_1}{2}
\]
\[
y = \frac{y_2 + y_1}{2}
\]
Area of Triangle
Area of triangle ABC of coordinates \( A(x_1, y_1) \) , \( B(x_2, y_2) \) and (\ C(x_3, y_3) \)
$A=\frac {1}{2}|x_1 (y_2-y_3 )+x_2 (y_3-y_1 )+x_3 (y_1-y_2 )|$
Formula For Collinearity
For point A, B and C to be collinear, the value of A should be zero
$|x_1 (y_2-y_3 )+x_2 (y_3-y_1 )+x_3 (y_1-y_2 )|=0$
Definition of Slope of the Straight line
- If $\theta$ is the inclination of a line l, then $\tan \theta$ is called the slope or gradient of the line l.
- The slope of a line whose inclination is 90° is not defined.
- The slope of a line is denoted by m.
- Thus, $m = \tan \theta$, $\theta \ne 90$
Slope Formula
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Angle Between Two Lines
If \( m_1 \) and \( m_2 \) are the slopes of two lines, then the angle ( $\theta$ ) between them is:
\[
\tan \theta =| \frac{m_2 – m_1}{1 + m_1 m_2}|
\]
If the lines are parallel, then $m_1=m_2$
if the lines are perpendicular $m_1m_2 =-1$
General Equation of Straight line
An equation of the form 0, $ax+ by+ c=0$ where a, b, c are constants and a, b are not simultaneously zero, always represents a straight line.
Various form Equation of Straight line
Point-slope form
For a line with slope ( m ) passing through the point \((x_1, y_1) \)
\[
y – y_1 = m(x – x_1)
\]
Two-point form
For a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \)
\[
y – y_1 = \frac{y_2 – y_1}{x_2 – x_1} (x – x_1)
\]
Slope Intercept Form
For a line with slope ( m ) and which cuts an intercept c on the y-axis
$y=mx +c $
Intercept Form
For a line intercepting the x-axis at ( a ) and the y-axis at ( b )
\[
\frac{x}{a} + \frac{y}{b} = 1
\]
Normal Form
For a line making an angle \( \omega \) with the positive x-axis and having a perpendicular distance ( p ) from the origin:
\[
x \cos \omega + y \sin \omega = p
\]
In normal form of equation of a straight line p is always taken as positive and ? is measured from positive direction of x-axis in anticlockwise direction between 0 and $2\pi$
Parametric Form
The equation of a straight line passing through the point $(x_1 ,y_1)$ and making an angle $\theta$ with the positive direction of x-axis is
$\frac {x-x_1}{\cos \theta}= \frac {y-y_1}{\sin \theta}= r$
Here $0 \leq \theta \leq \pi$
r is the distance of the point (x,y) from $(x_1,y_2)$
The coordinates of any point on the line at a distance r from the point $(x_1, y_1)$ can be taken as
$(x_1 + r cos \theta, y_1 + r sin \theta)$
Equations of straight lines passing through a given point and making an given angle with a given line
Equation of line passing through the point $(x_1,y_1)$ and making an angle $\alpha $ with the given line $y=m_1 x + c$ is given as
$y-y_1= \frac {m_1 \pm tan \alpha}{1 \mp m_1 tan \alpha} (x-x_1)$
Length of Perpendicular Point From a Line
The distance of a point from a line is the length of the perpendicular drawn from the point to the line. Let L : Ax + By + C = 0 be a line, whose distance from the point P (x1, y1) is d. Then d is given by
$d= \frac {|Ax_1 + By+1 + C|}{\sqrt {A^2 + B^2}}$
Distance between two Parallel Lines
For two parallel lines
$y=mx + c_1$
$y=mx+ c_2$
Distance between them is given by
$d= \frac {|c_1 -c_2|}{\sqrt {1+ m^2}}$
Position of two points with respect to given line
For points $(x_1,y_1)$ and $(x_2,y_2)$ with respect to given line $ax+ By + c =0$
if $\frac {ax_1 + by_1 + c}{ax_2 + by_2 + c} < 0$
Then points are opposite side of the line
if $\frac {ax_1 + by_1 + c}{ax_2 + by_2 + c} > 0$
Then the points are on same side
Family of Lines passing through intersection of two lines
if $L=0$ and $L^{‘}=0$ be the two line, then family of lines passing through the intersection of two lines is
$L+ \lambda L^{‘}=0$
Concurrency of Straight lines
For the three lines to be concurrent
$a_1 x + b_1 y + c_1=0$
$a_2 x + b_2 y + c_2=0$
$a_3 x + b_13y + c_3=0$
we should have
$ \begin{vmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{vmatrix}
$
or
$mL_1 + nL_2+ kl_3=0$ where m,n and k are constant which are non zero at the same time
Equations of the Bisectors of the angles between the lines
The equations of the bisectors of the angles between the lines $a_1 x + b_1 y + c_1=0$ and $a_2 x + b_2 y + c_2=0$ is
$\frac {a_1 x + b_1 y +c_1}{\sqrt {a_1^2 + b_1^2} }=\pm \frac {a_2 x + b_2 y +c_2}{\sqrt {a_2^2 + b_2^2} }
How to Find the equation of bisector of obtuse and acute angle
Let the line be such that $a_1 x + b_1 y + c_1=0$ and $a_2 x + b_2 y + c_2=0$ and $c_1 >0$ and $c_2> 0$
Then the equation of bisector containing Origin
$\frac {a_1 x + b_1 y +c_1}{\sqrt {a_1^2 + b_1^2} }= \frac {a_2 x + b_2 y +c_2}{\sqrt {a_2^2 + b_2^2} }$
And the other line is
$\frac {a_1 x + b_1 y +c_1}{\sqrt {a_1^2 + b_1^2} }= – \frac {a_2 x + b_2 y +c_2}{\sqrt {a_2^2 + b_2^2} }$
Now
If $a_1a_2 + b_1b_2 >0$ then the origin lies in obtuse angle i.e., the bisector containing origin is obtuse angle bisector and if $a_1a_2 + b_1b_2 <0$ then the origin lies in acute angle i.e., the bisector containing origin is acute angle bisector.
Foot of The Perpendicular from a point on Line
The co-ordinate of foot of perpendicular $x_2 ,y_2$ of a given point $x_1,y_1$ on the line $ax+ by+ c=0$ are given by
$\frac {x_2 -x_1}{a} = \frac {y_2 -y_1}{b} = – \frac {ax_1 + by_1 + c}{a^2 + b^2}$
Image of a point on a Line
The co-ordinate of image $x_2 ,y_2$ of a given point $x_1,y_1$ on the line $ax+ by+ c=0$ are given by
$\frac {x_2 -x_1}{a} = \frac {y_2 -y_1}{b} = -2 \frac {ax_1 + by_1 + c}{a^2 + b^2}$
Standard Points of the Triangle
Centroid Of the Triangle
- It is the concurrent point of the Median of the triangle
- Centroid divides the median in the ratio 2: 1
- Coordinates are given by
$x= \frac {x_1 + x_2 + x_3}{3}$ and $y= \frac {y_1+ y_2 + y_3}{3}$
Incenter of the Triangle
- It is the concurrent point of the angle bisectors
- Coordinates are given by
$x=\frac {ax_1 + bx_2 + cx_3}{a+ b+ c}$ and $y=\frac {ay_1 + by_2 + cy_3}{a+ b+c}$
Here a, b, c are the sides of the triangles
Circumcenter of the Triangle
- It is the concurrent point of the perpendicular bisectors of the sides of the triangles.
- If A, B, C are the angles of the $\Delta ABC$ and Vertices are \( A(x_1, y_1) \) , \( B(x_2, y_2) \) and (\ C(x_3, y_3) \), then circumcenter point is given as
$x=\frac {x_1 sin2A + x_2 sin2B + x_3 sin2C}{sin2A + sin2B+ sin2C}$ and $y=\frac {y_1 sin2A + y_2 sin2B+ y_3 sin2C}{sin2A + sin2B+ sin2C}$
Orthocenter of the Triangle
- The orthocenter of a triangle is the point of intersection of altitudes
- For an acute triangle, it lies inside the triangle.
- For an obtuse triangle, it lies outside of the triangle.
- If A, B, C are the angles of the $\Delta ABC$ and Vertices are \( A(x_1, y_1) \) , \( B(x_2, y_2) \) and (\ C(x_3, y_3) \), then orthocenter point is given as
$x=\frac {x_1 Tan A + x_2 Tan B + x_3 Tan C}{Tan A + Tan B+ Tan C}$ and $y=\frac {y_1 Tan A + y_2 Tan B+ y_3 Tan C}{Tan A + Tan B+ Tan C}$
Important Points
- In right angled the right angled vertex is the orthocenter of triangle and mid-point of hypotenuse is circumcenter
- In a equilateral Triangle, Centroid, Incenter, Circumcenter and Orthocenter coincides