Current Affairs JEE Main & Advanced

Equation of a Circle in Some Special Cases

(1) If centre of the circle is \[(h,\,k)\] and it passes through origin then its equation is \[{{(x-h)}^{2}}+{{(y-k)}^{2}}={{h}^{2}}+{{k}^{2}}\]\[\Rightarrow \,\,{{x}^{2}}+{{y}^{2}}\] \[-2hx-2ky=0\]. (2) If the circle touches x-axis then its equation is \[{{(x\pm h)}^{2}}+{{(y\pm k)}^{2}}={{k}^{2}}\]. (Four cases)

(3) If the circle touches y-axis then its equation is \[{{(x\pm h)}^{2}}+{{(y\pm k)}^{2}}={{h}^{2}}\]. (Four cases)

(4) If the circle touches both the axes then its equation is \[{{(x\pm r)}^{2}}+{{(y\pm r)}^{2}}={{r}^{2}}\] . (Four cases)

(5) If the circle touches x- axis at origin then its equation is \[{{x}^{2}}+{{(y\pm k)}^{2}}={{k}^{2}}\] \[\Rightarrow \,\,{{x}^{2}}+{{y}^{2}}\pm 2ky=0\]. (Two cases)

(6) If the circle touches y-axis at origin, the equation of circle is \[{{(x\pm h)}^{2}}+{{y}^{2}}={{h}^{2}}\]\[\Rightarrow \,\,{{x}^{2}}+{{y}^{2}}\pm 2xh=0\]. (Two cases)

(7) If the circle passes through origin and cut intercepts \[a\] and \[b\] on axes, the equation of circle is \[{{x}^{2}}+{{y}^{2}}-ax-by=0\] and centre is \[C(a/2,\,\,b/2)\]. (Four cases)

Catgeory: JEE Main & Advanced

Standard Forms of Equation of a Circle

(1) General equation of a circle : The general equation of a circle is \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] where \[g,f,c\] are constant. (i) Centre of the circle is\[(g,\text{ }f)\]. i.e., (\[-\frac{1}{2}\] coefficient of \[x,\,\,-\frac{1}{2}\] coefficient of \[y\]). (ii) Radius of the circle is \[\sqrt{{{g}^{2}}+{{f}^{2}}-c}\]. Nature of the circle (i) If \[{{g}^{2}}+{{f}^{2}}-c>0\], then the radius of the circle will be real. Hence, in this case, it is possible to draw a circle on a plane. (ii) If \[{{g}^{2}}+{{f}^{2}}-c=0\], then the radius of the circle will be zero. Such a circle is known as point circle. (iii) If \[{{g}^{2}}+{{f}^{2}}-c<0\], then the radius \[\sqrt{{{g}^{2}}+{{f}^{2}}-c}\] of the circle will be an imaginary number. Hence, in this case, it is not possible to draw a circle. The condition for the second degree equation to represent a circle : The general equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}\] \[+2gx+2fy+c=0\] represents a circle iff (i) \[a=b\ne 0\] (ii) \[h=0\] (iii) \[\Delta =abc+2hgf-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}\ne 0\] (iv) \[{{g}^{2}}+{{f}^{2}}-ac\ge 0\]

(2) Central form of equation of a circle : The equation of a circle having centre \[(h,\text{ }k)\] and radius \[r\] is \[{{(x-h)}^{2}}+{{(y-k)}^{2}}={{r}^{2}}\] If the centre is origin, then the equation of the circle is \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\] (3) Circle on a given diameter : The equation of the circle drawn on the straight line joining two given points \[({{x}_{1}},{{y}_{1}})\] and \[({{x}_{2}},\,{{y}_{2}})\] as diameter is

\[(x-{{x}_{1}})(x-{{x}_{2}})+(y-{{y}_{1}})(y-{{y}_{2}})=0\] (4) Parametric co-ordinates (i) The parametric co-ordinates of any point on the circle \[{{(x-h)}^{2}}+{{(y-k)}^{2}}={{r}^{2}}\] are given by\[(h+r\,\cos \,\theta ,\,k+r\,\sin \,\theta )\], \[(0\le \theta <2\pi )\]. In particular, co-ordinates of any point on the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\] are \[(r\,\cos \,\theta ,\,r\,\sin \,\theta ),\,\,(0\le \theta <2\pi )\]. (ii) The parametric co-ordinates of any point on the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] are \[x=-g+\sqrt{({{g}^{2}}+{{f}^{2}}-c})\,\,\cos \,\theta \] and \[y=-f+\sqrt{({{g}^{2}}+{{f}^{2}}-c)}\,\,\sin \,\theta ,\,\] \[(0\le \theta <2\pi )\] (5) Equation of a circle under given conditions (i) The equation of the circle through three non-collinear points \[A\,({{x}_{1}},\,{{y}_{1}}),\,B\,({{x}_{2}},\,{{y}_{2}}),\,\,C\,({{x}_{3}},\,{{y}_{3}})\,\]is \[\left| \,\,\begin{array}{*{35}{l}} {{x}^{2}}+{{y}^{2}} & x & y & 1 \\ x_{1}^{2}+y_{1}^{2} & {{x}_{1}} & {{y}_{1}} & 1 \\ x_{2}^{2}+y_{2}^{2} & {{x}_{2}} & {{y}_{2}} & 1 \\ x_{3}^{2}+y_{3}^{2} & {{x}_{3}} & {{y}_{3}} & 1 \\ \end{array}\,\, \right|\,=\,0\] (ii) From given three points taking any two as extremities of diameter of a circle \[S=0\] and equation of straight line passing through these two points is \[L=0\]. Then required equation of circle is \[S+\lambda L=0,\] where \[\lambda \] is a parameter, which can be found out by putting third point in the equation.

Catgeory: JEE Main & Advanced

Definition

A circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane always remains the same i.e., constant.

The fixed point is called the centre of the circle and the fixed distance is called the radius of the circle.

Catgeory: JEE Main & Advanced

Divisibility Problems

To show that an expression is divisible by an integer (i) If \[a,p,n,r\] are positive integers, then first of all we write \[{{a}^{pn+r}}={{a}^{pn}}.\,{{a}^{r}}={{({{a}^{p}})}^{n}}.\,{{a}^{r}}.\] (ii) If we have to show that the given expression is divisible by \[c\]. Then express, \[{{a}^{p}}=[1+({{a}^{p}}-1]\], if some power of \[({{a}^{p}}-1)\] has c as a factor. \[{{a}^{p}}=[2+({{a}^{p}}-2)]\], if some power of \[({{a}^{p}}-2)\] has c as a factor. \[{{a}^{p}}=[k+({{a}^{p}}-k)],\,\]if some power of \[({{a}^{p}}-k)\] has c as a factor.

Catgeory: JEE Main & Advanced

Second Principle of Mathematical Induction

The proof of proposition by mathematical induction consists of following steps : Step I : (Verification step) : Actual verification of the proposition for the starting value \[i\] and \[(i+1)\]. Step II : (Induction step) : Assuming the proposition to be true for \[k-1\] and \[k\] and then proving that it is true for the value \[k+1;\,\,k\,\,\ge \,\,i+1\]. Step III : (Generalization step) : Combining the above two steps. Let \[p(n)\] be a statement involving the natural number \[n\] such that (i) \[p(1)\] is true i.e. \[p(n)\]is true for \[n=1\] and (ii) \[p(m+1)\] is true, whenever \[p(n)\] is true for all \[n,\] where \[i\le n\le m\]. Then \[p(n)\]is true for all natural numbers. For \[a\ne b,\] The expression \[{{a}^{n}}-{{b}^{n}}\] is divisible by (a) \[a+b,\] if \[n\] is even. (b) \[a-b,\] if \[n\] is odd or even.

Catgeory: JEE Main & Advanced

First Principle of Mathematical Induction

The proof of proposition by mathematical induction consists of the following three steps : Step I : (Verification step) : Actual verification of the proposition for the starting value \[''i''\]. Step II : (Induction step) : Assuming the proposition to be true for \[''k'',\,\ge i\] and proving that it is true for the value \[(k+1)\] which is next higher integer. Step III : (Generalization step) : To combine the above two steps. Let \[p(n)\] be a statement involving the natural number n such that (i) \[p(1)\] is true i.e. \[p(n)\] is true for \[n=1\]. (ii) \[p(m+1)\] is true, whenever \[p(m)\] is true i.e. \[p(m)\] is true \[\Rightarrow \] \[p(m+1)\] is true. Then \[p(n)\] is true for all natural numbers \[n\].

Catgeory: JEE Main & Advanced

Some Important Points

(1) Pascal's Triangle more...

Catgeory: JEE Main & Advanced

Three / Four Consecutive Terms or Coefficients

(1) If consecutive coefficients are given: In this case divide consecutive coefficients pair wise. We get equations and then solve them. (2) If consecutive terms are given : In this case divide consecutive terms pair wise i.e. if four consecutive terms be \[{{T}_{r}},\,{{T}_{r+1}},\,{{T}_{r+2}},\,{{T}_{r+3}}\] then find \[\frac{{{T}_{r}}}{{{T}_{r+1}}},\,\frac{{{T}_{r+1}}}{{{T}_{r+2}}},\,\frac{{{T}_{r+2}}}{{{T}_{r+3}}}\] \[\Rightarrow \] \[{{\lambda }_{1}},\,{{\lambda }_{2}},\,{{\lambda }_{3}}\] (say) then divide \[{{\lambda }_{1}}\] by \[{{\lambda }_{2}}\] and \[{{\lambda }_{2}}\] by \[{{\lambda }_{3}}\] and solve.

Catgeory: JEE Main & Advanced

Binomial Theorem for any Index

Statement : \[{{(1+x)}^{n}}=1+nx+\frac{n(n-1){{x}^{2}}}{2!}+\frac{n(n-1)\,(n-2)}{3!}{{x}^{3}}+....\]\[+\frac{n(n-1)\,......(n-r+1)}{r!}{{x}^{r}}+...\text{terms up to }\infty \] when \[n\] is a negative integer or a fraction, where \[-1<x<1\], otherwise expansion will not be possible. If first term is not 1, then make first term unity in the following way, \[{{(x+y)}^{n}}={{x}^{n}}{{\left[ 1+\frac{y}{x} \right]}^{n}}\], if \[\left| \,\frac{y}{x}\, \right|<1\]. General term : \[{{T}_{r+1}}=\frac{n(n-1)(n-2)......(n-r+1)}{r!}{{x}^{r}}\] Some important expansions (i) \[{{(1+x)}^{n}}=1+nx+\frac{n(n-1)}{2!}{{x}^{2}}+\] \[.......+\frac{n(n-1)\,(n-2)......(n-r+1)}{r!}{{x}^{r}}+......\] (ii) \[{{(1-x)}^{n}}=1-nx+\frac{n(n-1)}{2!}{{x}^{2}}-.......\] \[+\frac{n(n-1)(n-2).....(n-r+1)}{r!}{{(-x)}^{r}}+.......\] (iii) \[{{(1-x)}^{-n}}=1+nx+\frac{n(n+1)}{2!}{{x}^{2}}+\frac{n(n+1)\,(n+2)}{3!}{{x}^{3}}+\]\[.....+\frac{n(n+1)......(n+r-1)}{r\,!}{{x}^{r}}+.....\] (iv) \[{{(1+x)}^{-n}}=1-nx+\frac{n(n+1)}{2!}{{x}^{2}}-\frac{n(n+1)(n+2)}{3\,!}{{x}^{3}}+\]\[.....+\frac{n(n+1)......(n+r-1)}{r!}{{(-x)}^{r}}+......\] (v) \[{{(1+x)}^{-1}}=1-x+{{x}^{2}}-{{x}^{3}}+.......\infty \] (vi) \[{{(1-x)}^{-1}}=1+x+{{x}^{2}}+{{x}^{3}}+.......\infty \] (vii) \[{{(1+x)}^{-2}}=1-2x+3{{x}^{2}}-4{{x}^{3}}+.......\infty \] (viii) \[{{(1-x)}^{-2}}=1+2x+3{{x}^{2}}+4{{x}^{3}}+.......\infty \] (ix) \[{{(1+x)}^{-3}}=1-3x+6{{x}^{2}}-...............\infty \] (x) \[{{(1-x)}^{-3}}=1+3x+6{{x}^{2}}+.................\infty \] Problems on approximation by the binomial theorem : We have \[{{(1+x)}^{n}}=1+nx+\frac{n(n-1)}{2!}{{x}^{2}}+.......\]. If \[x\] is small compared with 1, we find that the values of \[{{x}^{2}},{{x}^{3}},{{x}^{4}},.......\] become smaller and smaller. \[\therefore \] The terms in the above expansion become smaller and smaller. If \[x\] is very small compared with 1, we might take 1 as a first approximation to the value of \[{{(1+x)}^{n}}\] or \[1+nx\] as a second approximation.

Catgeory: JEE Main & Advanced

Multinomial Theorem (for Positive Integral Index)

If \[n\] is positive integer and \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,....{{a}_{n}}\in C\], then \[{{({{a}_{1}}+{{a}_{2}}+{{a}_{3}}+...+{{a}_{m}})}^{n}}=\sum{\frac{n!}{{{n}_{1}}!\,{{n}_{2}}!{{n}_{3}}!...{{n}_{m}}!}a_{1}^{{{n}_{1}}}.a_{2}^{{{n}_{2}}}}...a_{m}^{{{n}_{m}}}\], where \[{{n}_{1}},\,{{n}_{2}},\,{{n}_{3}},.....{{n}_{m}}\] are all non-negative integers subject to the condition, \[{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....{{n}_{m}}=n\]. (1) The coefficient of \[a_{1}^{{{n}_{1}}}.a_{2}^{{{n}_{2}}}.....a_{m}^{{{n}_{m}}}\] in the expansion of \[{{({{a}_{1}}+{{a}_{2}}+{{a}_{3}}+....{{a}_{m}})}^{n}}\] is \[\frac{n!}{{{n}_{1}}!{{n}_{2}}!{{n}_{3}}!....{{n}_{m}}!}\] (2) The greatest coefficient in the expansion of \[{{({{a}_{1}}+{{a}_{2}}+{{a}_{3}}+....{{a}_{m}})}^{n}}\] is \[\frac{n!}{{{(q!)}^{m-r}}{{[(q+1)!]}^{r}}}\] where \[q\] is the quotient and \[r\] is the remainder when \[n\]is divided by \[m\]. (3) If \[n\] is \[+ve\] integer and \[{{a}_{1}},\,{{a}_{2}},.....{{a}_{m}}\in C\] \[a_{1}^{{{n}_{1}}}\,.\,a_{2}^{{{n}_{2}}}\,.........a_{m}^{{{n}_{m}}}\] then coefficient of \[{{x}^{r}}\] in the expansion of \[{{({{a}_{1}}+{{a}_{2}}x+.....{{a}_{m}}{{x}^{m-1}})}^{n}}\] is \[\sum{\frac{n!}{{{n}_{1}}!{{n}_{2}}!{{n}_{3}}!.....{{n}_{m}}!}}\] where \[{{n}_{1}},\,{{n}_{2}},.....{{n}_{m}}\] are all non-negative integers subject to the condition: \[{{n}_{1}}+{{n}_{2}}+.....{{n}_{m}}=n\] and \[{{n}_{2}}+2{{n}_{3}}+3{{n}_{4}}+....+(m-1){{n}_{m}}=r\] (4) The number of distinct or dissimilar terms in the multinomial expansion \[{{({{a}_{1}}+{{a}_{2}}+{{a}_{3}}+....{{a}_{m}})}^{n}}\] is \[^{n+m-1}{{C}_{m-1}}\].

Catgeory: JEE Main & Advanced