Current Affairs JEE Main & Advanced

(1) Equation of plane passing through a given point : Equation of plane passing through the point \[({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] is \[A(x-{{x}_{1}})+B(y-{{y}_{1}})+C(z-{{z}_{1}})=0\], where \[A,\,\,B\] and \[C\] are d.r.’s of normal to the plane.     (2) Equation of plane through three points : The equation of plane passing through three non-collinear points \[({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\], \[({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] and \[({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})\] is \[\left| \,\begin{matrix} x & y & z & 1  \\ {{x}_{1}} & {{y}_{1}} & {{z}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & {{z}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & {{z}_{3}} & 1  \\ \end{matrix}\, \right|=0\] or \[\left| \,\begin{matrix} x-{{x}_{1}} & y-{{y}_{1}} & z-{{z}_{1}}  \\ {{x}_{2}}-{{x}_{1}} & {{y}_{2}}-{{y}_{1}} & {{z}_{2}}-{{z}_{1}}  \\ {{x}_{3}}-{{x}_{1}} & {{y}_{3}}-{{y}_{1}} & {{z}_{3}}-{{z}_{1}}  \\ \end{matrix}\, \right|=0\].

(1) Area of a triangle : The area of a triangle ABC with vertices \[A({{x}_{1}},{{y}_{1}}),\,\,B\text{ }({{x}_{2}},{{y}_{2}})\] and \[C({{x}_{3}},{{y}_{3}})\]. The area of triangle ABC is denoted by \['\Delta '\]and is given as       \[\Delta =\frac{1}{2}\left| \begin{matrix} {{x}_{1}} & {{y}_{1}} & 1  \\    {{x}_{2}} & {{y}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & 1  \\ \end{matrix} \right|\]\[=\frac{1}{2}\left| \text{ }({{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}})\text{ } \right|\]     In equilateral triangle     (i) Having sides a, area is \[\frac{\sqrt{3}}{4}{{a}^{2}}\].     (ii) Having length of perpendicular as 'p' area is \[\frac{({{p}^{2}})}{\sqrt{3}}\] .     (2) Collinear points : Three points \[A({{x}_{1}},{{y}_{1}}),\,\,B({{x}_{2}},{{y}_{2}}),\,C({{x}_{3}},{{y}_{3}})\] are collinear.  If area of triangle is zero, then     (i)  \[\Delta =0\]  \[\Rightarrow \]  \[\frac{1}{2}\left| \begin{matrix} {{x}_{1}} & {{y}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & 1  \\ \end{matrix} \right|=0\] \[\Rightarrow \] \[\left| \begin{matrix} {{x}_{1}} & {{y}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & 1  \\ {{x}_{3}} & {{y}_{3}} & 1  \\ \end{matrix} \right|=0\]     (ii) \[AB+BC=AC\] or \[AC+BC=AB\] or \[AC+AB=BC\]     (3) Area of a quadrilateral : If \[({{x}_{1}},{{y}_{1}}),\,({{x}_{2}},{{y}_{2}}),\,\,({{x}_{3}},{{y}_{3}})\] and \[({{x}_{4}},{{y}_{4}})\] are vertices of a quadrilateral, then its area   \[=\frac{1}{2}[({{x}_{1}}{{y}_{2}}-{{x}_{2}}{{y}_{1}})+({{x}_{2}}{{y}_{3}}-{{x}_{3}}{{y}_{2}})+({{x}_{3}}{{y}_{4}}-{{x}_{4}}{{y}_{3}})+({{x}_{4}}{{y}_{1}}-{{x}_{1}}{{y}_{4}})]\]     (4) Area of polygon : The area of polygon whose vertices are \[({{x}_{1}},{{y}_{1}}),({{x}_{2}},{{y}_{2}}),({{x}_{3}},{{y}_{3}}),....({{x}_{n,}}{{y}_{n}})\] is     \[=\,\frac{1}{2}|\{({{x}_{1}}{{y}_{2}}-{{x}_{2}}{{y}_{1}})+({{x}_{2}}{{y}_{3}}-{{x}_{3}}{{y}_{2}})+....+({{x}_{n}}{{y}_{1}}-{{x}_{1}}{{y}_{n}})\}|\]                     Or   Stair method : Repeat first co-ordinates one time in last for down arrow use positive sign and for up arrow use negative sign.     \[\therefore \] Area of polygon     \[=\frac{1}{2}|\{({{x}_{1}}{{y}_{2}}+{{x}_{2}}{{y}_{3}}+....+{{x}_{n}}{{y}_{1}})-({{y}_{1}}{{x}_{2}}+{{y}_{2}}{{x}_{3}}+....+{{y}_{n}}{{x}_{1}})\}|\]

If point \[P(x,y,z)\] moves according to certain rule, then it may lie in a 3-D region on a surface or on a line or it may simply be a point. Whatever we get, as the region of P after applying the rule, is called locus of P. Let us discuss about the plane or curved surface. If Q be any other point on it’s locus and all points of the straight line PQ lie on it, it is a plane. In other words if the straight line PQ, however small and in whatever direction it may be, lies completely on the locus, it is a plane, otherwise any curved surface.   (1) General equation of plane : Every equation of first degree of the form \[Ax+By+Cz+D=0\] represents the equation of a plane. The coefficients of \[x,\,\,y\] and z i.e., \[A,\,\,B,\,\,C\] are the direction ratios of the normal to the plane.     (2) Equation of co-ordinate planes : \[XOY\]-plane : \[z=\text{ }0,\] \[YOZ\]-plane : \[x=0,\] \[ZOX\]-plane : \[y=0\]     (3) Equation of plane in various forms :   (i) Intercept form : If the plane cuts the intercepts of length \[a,\,\,b,\,\,c\] on co-ordinate axes, then its equation is \[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\].   (ii) Normal form : Normal form of the equation of plane is \[lx+my+nz=p\], where \[l,\,\,m,\,\,c\] are the d.c.’s of the normal to the plane and \[p\] is the length of perpendicular from the origin.     (4) Equation of plane in particular cases : Equation of plane through the origin is given by \[Ax+By+Cz=0\].     i.e, if \[D=0\], then the plane passes through the origin.     (5) Equation of plane parallel to co-ordinate planes or perpendicular to co-ordinate axes     (i) Equation of plane parallel to \[YOZ\]-plane (or perpendicular to x-axis) and at a distance \['a'\] from it is \[x=a\].     (ii) Equation of plane parallel to \[ZOX\]-plane (or perpendicular to y-axis) and at a distance \['b'\] from it is \[y=b\].     (iii) Equation of plane parallel to \[XOY\]-plane (or perpendicular to z-axis) and at a distance \['c'\] from it is \[z=c\].     (6) Equation of plane perpendicular to co-ordinate planes or parallel to co-ordinate axes     (i) Equation of plane perpendicular to \[YOZ\]-plane or parallel to x-axis is \[By+Cz+D=0\].     (ii) Equation of plane perpendicular to \[ZOX\]-plane or parallel to y-axis is \[Ax+Cz+D=0\].     (iii) Equation of plane perpendicular to \[XOY\]-plane or parallel to z-axis is \[Ax+By+D=0\].     (7) Equation of plane parallel to a given plane : Plane parallel to a given plane \[ax+by+cz+d=0\] is \[ax+by+cz+{d}'=0\], i.e. only constant term is changed.       (8) Equation of plane passing through the intersection of two planes : Equation of plane through the intersection of two planes \[P={{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}=0\] and \[Q={{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}=0\] is \[P+\lambda Q=0\], where \[\lambda \] is the parameter.  

  (1) Skew lines : Two straight lines in space which are neither parallel nor intersecting are called skew lines.     Thus, the skew lines are those lines which do not lie in the same plane.         (2) Line of shortest distance : If \[{{l}_{1}}\] and \[{{l}_{2}}\] are two skew lines, then the straight line which is perpendicular to each of these two non-intersecting lines is called the “Line of shortest distance.”     There is one and only one line perpendicular to each of lines \[{{l}_{1}}\] and \[{{l}_{2}}\].     (3) Shortest distance between two skew lines     Let two skew lines be, \[\frac{x-{{x}_{1}}}{{{l}_{1}}}=\frac{y-{{y}_{1}}}{{{m}_{1}}}=\frac{z-{{z}_{1}}}{{{n}_{1}}}\] and \[\frac{x-{{x}_{2}}}{{{l}_{2}}}=\frac{y-{{y}_{2}}}{{{m}_{2}}}=\frac{z-{{z}_{2}}}{{{n}_{2}}}\]     Therefore, the shortest distance between the lines is given by  \[d=\frac{\left| \,\begin{matrix} {{x}_{2}}-{{x}_{1}} & {{y}_{2}}-{{y}_{1}} & {{z}_{2}}-{{z}_{1}}  \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}}  \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}}  \\ \end{matrix}\, \right|}{\sqrt{{{({{m}_{1}}{{n}_{2}}-{{m}_{2}}{{n}_{1}})}^{2}}+{{({{n}_{1}}{{l}_{2}}-{{l}_{1}}{{n}_{2}})}^{2}}+{{({{l}_{1}}{{m}_{2}}-{{m}_{1}}{{l}_{2}})}^{2}}}}\].    

Foot of perpendicular from a point \[A(\alpha ,\,\,\beta ,\,\,\gamma )\]to the line  \[\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}\] : If P be the foot of perpendicular, then P is \[(lr+{{x}_{1}},\,mr+{{y}_{1}},\,nr+{{z}_{1}})\]. Find the direction ratios of AP and apply the condition of perpendicularity of AP and the given line. This will give the value of r and hence the point P, which is foot of perpendicular.     Length and equation of perpendicular : The length of the perpendicular is the distance AP and its equation is the line joining two known points A and P.     The length of the perpendicular is the perpendicular distance of given point from that line.     Reflection or image of a point in a straight line : If the perpendicular PL from point P on the given line be produced to Q such that \[PL=QL,\] then Q is known as the image or reflection of P in the given line. Also, L is the foot of the perpendicular or the projection of P on the line.      

Determine whether two lines intersect or not. In case they intersect, the following algorithm is used to find their point of intersection.     Algorithm:     Let the two lines be  \[\frac{x-{{x}_{1}}}{{{a}_{1}}}=\frac{y-{{y}_{1}}}{{{b}_{1}}}=\frac{z-{{z}_{1}}}{{{c}_{1}}}\]             …..(i)     and  \[\frac{x-{{x}_{2}}}{{{a}_{2}}}=\frac{y-{{y}_{2}}}{{{b}_{2}}}=\frac{z-{{z}_{2}}}{{{c}_{2}}}\]                        …..(ii)     Step I : Write the co-ordinates of general points on (i) and (ii). The co-ordinates of general points on (i) and (ii) are given by \[\frac{x-{{x}_{1}}}{{{a}_{1}}}=\frac{y-{{y}_{1}}}{{{b}_{1}}}=\frac{z-{{z}_{1}}}{{{c}_{1}}}=\lambda \] and \[\frac{x-{{x}_{2}}}{{{a}_{2}}}=\frac{y-{{y}_{2}}}{{{b}_{2}}}=\frac{z-{{z}_{2}}}{{{c}_{2}}}=\mu \] respectively.   i.e., \[({{a}_{1}}\lambda +{{x}_{1}},\,{{b}_{1}}\lambda +{{y}_{1}}+{{c}_{1}}\lambda +{{z}_{1}})\]and \[({{a}_{2}}\mu +{{x}_{2}},\,{{b}_{2}}\mu +{{y}_{2}},\,{{c}_{2}}\mu +{{z}_{2}})\].     Step II : If the lines (i) and (ii) intersect, then they have a common point. \[{{a}_{1}}\lambda +{{x}_{1}}={{a}_{2}}\mu +{{x}_{2}},\,{{b}_{1}}\lambda +{{y}_{1}}={{b}_{2}}\mu +{{y}_{2}}\] and \[{{c}_{1}}\lambda +{{z}_{1}}={{c}_{2}}\mu +{{z}_{2}}\].     Step III : Solve any two of the equations in \[\lambda \] and \[\mu \] obtained in step II. If the values of l and m satisfy the third equation, then the lines (i) and (ii) intersect, otherwise they do not intersect.     Step IV : To obtain the co-ordinates of the point of intersection, substitute the value of \[\lambda \] (or \[\mu \]) in the co-ordinates of general point (s) obtained in step I.

The unsymmetrical form of a line \[ax+by+cz+d=0,\] \[\,{a}'x+{b}'y+{c}'z+{d}'=0\] can be changed to symmetrical form as follows : \[\frac{x-\frac{b{d}'-{b}'d}{a{b}'-{a}'b}}{b{c}'-{b}'c}=\frac{y-\frac{d{a}'-{d}'a}{a{b}'-{a}'b}}{c{a}'-{c}'a}=\frac{z}{a{b}'-{a}'b}\].

(1) Centroid of a triangle : The centroid of a triangle is the point of intersection of its medians. The centroid divides the medians in the ratio 2 : 1   (vertex : base)         If \[A({{x}_{1}},{{y}_{1}})\], \[B({{x}_{2}},{{y}_{2}})\] and \[C({{x}_{3}},{{y}_{3}})\] are the vertices of a triangle. If G be the centroid upon one of the median (say) AD, then AG : GD = 2 : 1     \[\Rightarrow \] Co-ordinate of G are \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\]     (2) Circumcentre : The circumcentre of a triangle is the point of intersection of the perpendicular bisectors of the sides of a triangle. It is the centre of the circle which passes through the vertices of the triangle and so its distance from the vertices of the triangle is the same and this distance is known as the circum-radius of the triangle.         Let vertices A, B, C of the triangle ABC be \[({{x}_{1}},{{y}_{1}}),({{x}_{2}},{{y}_{2}})\] and \[({{x}_{3}},{{y}_{3}})\]and let circumcentre be \[O(x,\,\,y)\] and then \[(x,\,\,y)\] can be found by solving \[{{(OA)}^{2}}={{(OB)}^{2}}={{(OC)}^{2}}\]     i.e., \[{{(x-{{x}_{1}})}^{2}}+{{(y-{{y}_{1}})}^{2}}={{(x-{{x}_{2}})}^{2}}+{{(y-{{y}_{2}})}^{2}}\]\[={{(x-{{x}_{3}})}^{2}}+{{(y-{{y}_{3}})}^{2}}\]                  If a triangle is right angle, then its circumcentre is the mid point of hypotenuse. If angles of triangle i.e., A, B, C and vertices of triangle \[A({{x}_{1}},{{y}_{1}}),B({{x}_{2}},{{y}_{2}})\] and \[C\,({{x}_{3}},{{y}_{3}})\] are given, then circumcentre of the triangle ABC is     \[\left( \frac{{{x}_{1}}\sin 2A+{{x}_{2}}\sin 2B+{{x}_{3}}\sin 2C}{\sin 2A+\sin 2B+\sin 2C} \right.,\left. \frac{{{y}_{1}}\sin 2A+{{y}_{2}}\sin 2B+{{y}_{3}}\sin 2C}{\sin 2A+\sin 2B+\sin 2C} \right)\]     (3) Incentre : The incentre of a triangle is the point of intersection of internal bisector of the angles. Also it is a centre of a circle touching all the sides of a triangle.         Co-ordinates of incentre     \[\left( \frac{a{{x}_{1}}+b{{x}_{2}}+c{{x}_{3}}}{a+b+c},\frac{a{{y}_{1}}+b{{y}_{2}}+c{{y}_{3}}}{a+b+c} \right)\]     where a, b, c are the sides of triangle ABC.     (4) Excircle : A circle touches one side outside the triangle and other two extended sides then circle is known as excircle. Let ABC be a triangle then there are three excircles with three excentres. Let \[{{I}_{1}},{{I}_{2}},{{I}_{3}}\] be the centres of ex-circles opposite to vertices A, B and C respectively. If vertices of triangle are \[A({{x}_{1}},{{y}_{1}}),\] \[B({{x}_{2}},{{y}_{2}})\] and \[C\,({{x}_{3}},{{y}_{3}})\], then             \[{{I}_{1}}\equiv \left( \frac{-a{{x}_{1}}+b{{x}_{2}}+c{{x}_{3}}}{-a+b+c},\frac{-a{{y}_{1}}+b{{y}_{2}}+c{{y}_{3}}}{-a+b+c} \right)\],     \[{{I}_{2}}\equiv \left( \frac{a{{x}_{1}}-b{{x}_{2}}+c{{x}_{3}}}{a-b+c},\frac{a{{y}_{1}}-b{{y}_{2}}+c{{y}_{3}}}{a-b+c} \right)\],     \[{{I}_{3}}\equiv \left( \frac{a{{x}_{1}}+b{{x}_{2}}-c{{x}_{3}}}{a+b-c},\frac{a{{y}_{1}}+b{{y}_{2}}-c{{y}_{3}}}{a+b-c} \right)\]     Angle bisector divides the opposite sides in the ratio of remaining sides e.g. \[\frac{BD}{DC}=\frac{AB}{AC}=\frac{c}{b}\].     Incentre divides the angle bisectors in the ratio \[(b+c):a,\text{  }(c+a):b\] and \[(a+b):c\].     Excentre : Point of intersection of one internal angle bisector and other two external angle bisector is called as excentre. There are three excentres in a triangle. Co-ordinate of each can more...

Every equation of the first degree represents a plane. Two equations of the first degree are satisfied by the co-ordinates of every point on the line of intersection of the planes represented by them.     Therefore, the two equations of that line \[ax+by+cz+d=0\] and \[{a}'x+{b}'y+{c}'z+{d}'=0\] together represent a straight line.     (1) Equation of a line passing through a given point     Cartesian equation of a straight line passing through a fixed point \[({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and having direction ratios \[a,b,c\] is \[\frac{x-{{x}_{1}}}{a}=\frac{y-{{y}_{1}}}{b}=\frac{z-{{z}_{1}}}{c}\].     (2) Equation of line passing through two given points     If \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),\,B({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] be two given points, the equations to the line AB are \[\frac{x-{{x}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{y-{{y}_{1}}}{{{y}_{2}}-{{y}_{1}}}=\frac{z-{{z}_{1}}}{{{z}_{2}}-{{z}_{1}}}\].

Let \[\theta \]  be the angle between two straight lines AB and AC whose direction  cosines are \[{{l}_{1}},\,{{m}_{1}},\,{{n}_{1}}\] and \[{{l}_{2}},\,{{m}_{2}},\,{{n}_{2}}\] respectively, is given by\[\cos \theta ={{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}\].     If direction ratios of two lines \[{{a}_{1}},\,{{b}_{1}},\,{{c}_{1}}\] and \[{{a}_{2}},\,{{b}_{2}},\,{{c}_{2}}\] are given, then angle between two lines is given by     \[\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}.\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\].     Particular results: We have, \[{{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta \]     \[=(l_{1}^{2}+m_{1}^{2}+n_{1}^{2})(l_{2}^{2}+m_{2}^{2}+n_{2}^{2})-{{({{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}})}^{2}}\]     \[={{({{l}_{1}}{{m}_{2}}-{{l}_{2}}{{m}_{1}})}^{2}}+{{({{m}_{1}}{{n}_{2}}-{{m}_{2}}{{n}_{1}})}^{2}}+{{({{n}_{1}}{{l}_{2}}-{{n}_{2}}{{l}_{1}})}^{2}}\]     \[\Rightarrow \] \[\sin \theta =\pm \sqrt{\sum {{({{l}_{1}}{{m}_{2}}-{{l}_{2}}{{m}_{1}})}^{2}}}\], which is known as Lagrange’s identity.     The value of \[\sin \,\theta \] can easily be obtained by,    \[\sin \theta =\sqrt{{{\left| \begin{matrix} {{l}_{1}} & {{m}_{1}}  \\ {{l}_{2}} & {{m}_{2}}  \\ \end{matrix} \right|}^{2}}+{{\left| \begin{matrix} {{m}_{1}} & {{n}_{1}}  \\ {{n}_{2}} & {{n}_{2}}  \\ \end{matrix} \right|}^{2}}+{{\left| \begin{matrix} {{n}_{1}} & {{l}_{1}}  \\ {{n}_{2}} & {{l}_{2}}  \\ \end{matrix} \right|}^{2}}}\]   If \[{{a}_{1}},\,{{b}_{1}},\,{{c}_{1}}\] and \[{{a}_{2}},\,{{b}_{2}},\,{{c}_{2}}\] are d.r.’s of two given lines, then angle \[\theta \] between them is given by \[\sin \theta =\frac{\sqrt{\sum {{({{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}})}^{2}}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\]     Condition of perpendicularity : If the given lines are perpendicular, then \[\theta =90{}^\circ \] i.e., \[\cos \theta =0\]     \[\Rightarrow \] \[{{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}=0\] or \[{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}=0\]   Condition of parallelism : If the given lines are parallel, then \[\theta ={{0}^{o}}\] i.e.,  \[\sin \,\theta =0\Rightarrow \frac{{{l}_{1}}}{{{l}_{2}}}=\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{{{n}_{1}}}{{{n}_{2}}}\].   Similarly, \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\].