# Current Affairs JEE Main & Advanced

#### Angle Between two Non-parallel Lines

If $\theta$  be the angle between the lines $y={{m}_{1}}x+{{c}_{1}}$ and $y={{m}_{2}}x+{{c}_{2}}$ and intersecting at A. Then, $\theta ={{\tan }^{-1}}\,\left| \frac{{{m}_{1}}-{{m}_{2}}}{1+{{m}_{1}}{{m}_{2}}} \right|$.   If $\theta$ is angle between two lines, then $\pi -\theta$ is also the angle between them.     (1) Angle between two straight lines when their equations are given : The angle q between the lines ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ is given by, $\tan \theta =\left| \,\frac{{{a}_{2}}{{b}_{1}}-{{a}_{1}}{{b}_{2}}}{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}}\, \right|$.     (2) Conditions for two lines to be coincident, parallel, perpendicular and intersecting : Two lines ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$  and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ are,     (a) Coincident, if  $\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}$                  (b) Parallel, if $\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}$     (c) Intersecting, if $\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}$     (d) Perpendicular, if  ${{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}=0$

#### General Equation of Lines Through the Intersection of Two Given Lines

If equation of two lines $P={{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ and $Q={{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$, then the equation of the lines passing through the point of intersection of these lines is $P+\lambda \,Q=0$ or ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}+$$\lambda ({{a}_{2}}x+{{b}_{2}}y+{{c}_{2}})=0$. Value of $\lambda$ is obtained with the help of the additional information given in the problem.

#### Point of Intersection of Two Lines

Point of intersection of two lines ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ is given by   $({x}',\,{y}')=\left( \frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}\,,\,\frac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}} \right)=\left( \frac{\left| \,\begin{matrix} {{b}_{1}} & {{b}_{2}} \\ {{c}_{1}} & {{c}_{2}} \\ \end{matrix}\, \right|}{\left| \,\begin{matrix} {{a}_{1}} & {{a}_{2}} \\ {{b}_{1}} & {{b}_{2}} \\ \end{matrix}\, \right|}\,\,,\,\,\frac{\left| \,\begin{matrix} {{c}_{1}} & {{c}_{2}} \\ {{a}_{1}} & {{a}_{2}} \\ \end{matrix}\, \right|}{\left| \,\begin{matrix} {{a}_{1}} & {{a}_{2}} \\ {{b}_{1}} & {{b}_{2}} \\ \end{matrix}\, \right|} \right)$

#### General Equation of a Straight Line and its Transformation in Standard Forms

General form of equation of a line is $ax+by+c=0$, its     (1) Slope intercept form: $y=-\frac{a}{b}x-\frac{c}{b}$, slope $m=\frac{a}{b}$ and intercept on y-axis is, $C=-\frac{c}{b}$.     (2) Intercept form : $\frac{x}{-c/a}+\frac{y}{-c/b}=1$, $x$ intercept is $=\left( -\frac{c}{a} \right)$ and y intercept is $=\left( -\frac{c}{b} \right)$.     (3) Normal form : To change the general form of a line into normal form, first take c to right hand side and make it positive, then divide the whole equation by $\sqrt{{{a}^{2}}+{{b}^{2}}}$ like     $-\frac{ax}{\sqrt{{{a}^{2}}+{{b}^{2}}}}-\frac{by}{\sqrt{{{a}^{2}}+{{b}^{2}}}}=\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}},$     where $\cos \alpha =-\frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}}},\,\,\sin \alpha =-\frac{b}{\sqrt{{{a}^{2}}+{{b}^{2}}}},\,\,p=\frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}}$

#### Equation of Parallel and Perpendicular Lines to a Given Line

(1) Equation of a line which is parallel to $ax+by+c=0$ is $ax+by+\lambda =0$.   (2) Equation of a line which is perpendicular to $ax+by+c=0$ is $bx-ay+\lambda =0$.   The value of $\lambda$ in both cases is obtained with the help of additional information given in the problem.   (3) If the equation of line be $a\sin \theta +b\cos \theta =c$, then line   (i) Parallel to it, $a\sin \theta +b\cos \theta =d$   (ii) Perpendicular to it, $a\sin \left( \frac{\pi }{2}+\theta \right)+b\cos \left( \frac{\pi }{2}+\theta \right)=d$.

#### Equations of Straight Line in Different Forms

(1) Slope form : Equation of a line through the origin and having slope $m$ is $y=mx$.     (2) One point form or Point slope form : Equation of a line through the point $({{x}_{1}},{{y}_{1}})\,$and having slope $m$ is $y-{{y}_{1}}=m\,(x-{{x}_{1}})$.     (3) Slope intercept form  : Equation of a line (non-vertical) with slope m and cutting off an intercept $c$ on the y-axis is $y=mx+c$.     The equation of a line with slope m and the x-intercept $d$ is $y=m(x-d)$.     (4) Intercept form : If a straight line cuts x-axis at A and the y-axis at B then OA and OB are known as the intercepts of the line on x-axis and y-axis respectively.     Then, equation of a straight line cutting off intercepts $a$ and $b$ on x–axis and y–axis respectively is $\frac{x}{a}+\frac{y}{b}=1$.     If given line is parallel to $X$ axis, then X-intercept is undefined.     If given line is parallel to $Y$ axis, then Y-intercept is undefined.     (5) Two point form: Equation of the line through the points  $A({{x}_{1}},{{y}_{1}})\,$  and $B({{x}_{2}},{{y}_{2}})$ is, $(y-{{y}_{1}})=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}(x-{{x}_{1}})$.       In the determinant form it is gives as $\left| \,\begin{matrix} x & y & 1 \\ {{x}_{1}} & {{y}_{1}} & 1 \\ {{x}_{2}} & {{y}_{2}} & 1 \\ \end{matrix}\, \right|=0$  is the equation of line.     (6) Normal or perpendicular form : The equation of the straight line upon which the length of the perpendicular from the origin is $p$ and this perpendicular makes an angle $\alpha$ with x-axis is $x\cos \alpha +y\sin \alpha =p$.     (7) Symmetrical or parametric or distance form of the line :  Equation of a line passing through $({{x}_{1}},{{y}_{1}})$ and making an angle q with the positive direction of x-axis is $\frac{x-{{x}_{1}}}{\cos \theta }=\frac{y-{{y}_{1}}}{\sin \theta }=\pm r$, where r is the distance between the point $P\,(x,\,\,y)$  and $A({{x}_{1}},{{y}_{1}})$.     The co-ordinates of any point on this line may be taken as $({{x}_{1}}\pm r\cos \theta ,{{y}_{1}}\pm r\sin \theta )$, known as parametric co-ordinates. $'r'$  is called the parameter.

#### Slope (Gradient) of a Line

The trigonometrical tangent of the angle that a line makes with the positive direction of the x-axis in anticlockwise sense is called the slope or gradient of the line. The slope of a line is generally denoted by $m$. Thus, $m=\tan \theta$.         (1) Slope of line parallel to x – axis is $m=\tan {{0}^{o}}=0$.     (2) Slope of line parallel to y – axis is $m=\tan {{90}^{o}}=\infty$.     (3) Slope of the line equally inclined with the axes is 1 or – 1.     (4) Slope of the line through the points $A({{x}_{1}},{{y}_{1}})$ and $B({{x}_{2}},{{y}_{2}})$ is  $\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ taken in the same order.     (5) Slope of the line$ax+by+c=0,b\ne 0$ is $-\frac{a}{b}$.     (6) Slope of two parallel lines are equal.     (7) If ${{m}_{1}}$ and ${{m}_{2}}$ be the slopes of two perpendicular lines, then ${{m}_{1}}.{{m}_{2}}=-1$.     (8) $m$ can be defined as $\tan \theta$ for $0\le \theta \le \pi$ and $\theta \ne \frac{\pi }{2}$.

#### Definition

The straight line is a curve such that every point on the line segment joining any two points on it lies on it. The simplest locus of a point in a plane is a straight line. A line is determined uniquely by any one of the following:     (1) Two different points (because we know the axiom that one and only one straight line passes through two given points).     (2) A point and a given direction.

#### Probability Theory

Numerical study of chances of occurrence of events is dealt in probability theory.     The theory of probability is applied in many diverse fields and the flexibility of the theory provides approximate tools for so great a variety of needs.

#### Introduction

The word trigonometry is derived from two greek words "trigonon" and "metron". The word "trigonon" means a triangle and the word "metron" means a measure. Hence the word trigonometry means the study of properties of triangles. This involves the measurement of angles and lengths.     The motion of any revolving line in a plane from its initial position (initial side) to the final position (terminal side) is called angle. The end point O about which the line rotates is called the vertex of the angle.

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