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The locus of the point of intersection of the tangents to the hyperbola at A and B is called the polar of the given point \[P\] with respect to the hyperbola and the point \[P\] is called the pole of the polar. The equation of the required polar with \[({{x}_{1}},\,{{y}_{1}})\] as its pole is  \[\frac{x{{x}_{1}}}{{{a}^{2}}}-\frac{y{{y}_{1}}}{{{b}^{2}}}=1\].             Pole of a given line: The pole of a given line \[lx+my+n=0\] with respect to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[({{x}_{1}},{{y}_{1}})=\]\[\left( -\frac{{{a}^{2}}l}{n},\,\frac{{{b}^{2}}m}{n} \right)\].   Properties of pole and polar   (i) If the polar of \[P({{x}_{1}},\,{{y}_{1}})\] passes through \[{{y}_{1}},{{y}_{2}},\,{{y}_{3}}\], then the polar of \[Q({{x}_{2}},{{y}_{2}})\] goes through \[P({{x}_{1}},\,{{y}_{1}})\] and such points are said to be conjugate points.   (ii) If the pole of a line \[lx+my+n=0\] lies on the another line \[4{{x}^{2}}-(4h-k)\,x-1=0\] then the pole of the second line will lie on the first and such lines are said to be conjugate lines.   (iii) Pole of a given line is same as point of intersection of tangents as its extremities.

The locus of the middle points of a system of parallel chords of a hyperbola is called a diameter and the point where the diameter intersects the hyperbola is called the vertex of the diameter.             Let \[y=mx+c\] a system of parallel chords to \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] for different chords then the equation of diameter of the hyperbola is \[y=\frac{{{b}^{2}}x}{{{a}^{2}}m},\] which is passing through (0, 0).     Conjugate diameter : Two diameters are said to be conjugate when each bisects all chords parallel to the others.     If \[y={{m}_{1}}x,\,\,y={{m}_{2}}x\]  be conjugate diameters, then \[{{m}_{1}}{{m}_{2}}=\frac{{{b}^{2}}}{{{a}^{2}}}\].

Let the tangent and normal at \[P({{x}_{1}},\,{{y}_{1}})\] meet the x-axis at A and B respectively.           Length of subtangent     \[AN=CN-CA={{x}_{1}}-\frac{{{a}^{2}}}{{{x}_{1}}}\].     Length of subnormal     \[BN=CB-CN=\frac{({{a}^{2}}+{{b}^{2}})}{{{a}^{2}}}{{x}_{1}}-{{x}_{1}}\]     =\[\frac{{{b}^{2}}}{{{a}^{2}}}{{x}_{1}}=({{e}^{2}}-1){{x}_{1}}\].

An asymptote to a curve is a straight line, at a finite distance from the origin, to which the tangent to a curve tends as the point of contact goes to infinity.     The equations of two asymptotes of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are \[{{x}^{2}}=4y\] or \[\frac{x}{a}\pm \frac{y}{b}=0\].     Some important points about asymptotes       (i) The combined equation of the asymptotes of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=0\].     (ii) When \[b=a\] i.e. the asymptotes of rectangular hyperbola \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\] are \[y=\pm x\], which are at right angles.     (iii) A hyperbola and its conjugate hyperbola have the same asymptotes.                 (iv) The equation of the pair of asymptotes differ the hyperbola and the conjugate hyperbola by the same constant only i.e., Hyperbola – Asymptotes = Asymptotes – Conjugated hyperbola or     \[\left( \frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}-1 \right)-\left( \frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}} \right)=\left( \frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}} \right)-\left( \frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}+1 \right)\].     (v) The asymptotes pass through the centre of the hyperbola.     (vi) The bisectors of the angles between the asymptotes are the coordinate axes.     (vii) The angle between the asymptotes of the hyperbola \[S=0\] i.e., \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[2{{\tan }^{-1}}\frac{b}{a}\] or 2\[{{\sec }^{-1}}e\].     (viii) Asymptotes are equally inclined to the axes of the hyperbola.

  (1) Definition : A hyperbola whose asymptotes are at right angles to each other is called a rectangular hyperbola. The eccentricity of rectangular hyperbola is always \[\sqrt{2}\].     The general equation of second degree represents a rectangular hyperbola if \[\Delta \ne 0,\,\,{{h}^{2}}>ab\] and  coefficient of \[{{x}^{2}}+\] coefficient of \[{{y}^{2}}=0\].     (2) Parametric co-ordinates of a point on the hyperbola \[\mathbf{XY=}{{\mathbf{c}}^{\mathbf{2}}}\] : If \[t\] is non–zero variable, the coordinates of any point on the rectangular hyperbola \[xy={{c}^{2}}\] can be written as \[(ct,c/t)\]. The point \[(ct,c/t)\] on the hyperbola \[xy={{c}^{2}}\] is generally referred as the point \['t'\].     For rectangular hyperbola the coordinates of foci are \[(\pm a\sqrt{2},\,0)\] and directrices are \[x=\pm a\sqrt{2}\].     For rectangular hyperbola \[xy={{c}^{2}},\]  the coordinates of foci are \[{{a}^{2}}{{l}^{2}}+{{b}^{2}}{{m}^{2}}={{n}^{2}}\] and directrices are \[x+y=\pm c\sqrt{2}\].     (3) Equation of the chord joining points \[{{\mathbf{t}}_{\mathbf{1}}}\] and \[{{\mathbf{t}}_{\mathbf{2}}}\] : The equation of the chord joining two points \[\left( c{{t}_{1}},\frac{c}{{{t}_{1}}} \right)\,\text{and}\,\left( c{{t}_{2}},\frac{c}{{{t}_{2}}} \right)\]  on the hyperbola \[xy={{c}^{2}}\] is \[y-\frac{c}{{{t}_{1}}}=\frac{\frac{c}{{{t}_{2}}}-\frac{c}{{{t}_{1}}}}{c{{t}_{2}}-c{{t}_{1}}}(x-c{{t}_{1}})\]     \[\Rightarrow x+y\,{{t}_{1}}{{t}_{2}}=c\,({{t}_{1}}+{{t}_{2}})\].     (4) Equation of tangent in different forms     (i) Point form : The equation of tangent at \[({{x}_{1}},{{y}_{1}})\] to the hyperbola \[xy={{c}^{2}}\] is \[x{{y}_{1}}+y{{x}_{1}}=2{{c}^{2}}\] or \[\frac{x}{{{x}_{1}}}+\frac{y}{{{y}_{_{1}}}}=2\].     (ii) Parametric form :  The equation of the tangent at \[\left( ct,\frac{c}{t} \right)\] to the hyperbola \[xy={{c}^{2}}\] is \[\frac{x}{t}+yt=2c\].On replacing \[{{x}_{1}}\] by \[ct\] and \[{{y}_{1}}\] by \[\frac{c}{t}\] on the equation of the tangent at \[({{x}_{1}},{{y}_{1}})\]     i.e., \[x{{y}_{1}}+y{{x}_{1}}=2{{c}^{2}}\] we get \[\frac{x}{t}+yt=2c\].     Point of intersection of tangents at \['{{t}_{1}}'\] and \['{{t}_{2}}'\] is \[\left( \frac{2c{{t}_{1}}{{t}_{2}}}{{{t}_{1}}+{{t}_{2}}},\,\frac{2c}{{{t}_{1}}+{{t}_{2}}} \right)\].     (5) Equation of the normal in different forms :     (i) Point form : The equation of the normal at \[({{x}_{1}},{{y}_{1}})\] to the hyperbola \[xy={{c}^{2}}\] is \[x{{x}_{1}}-y{{y}_{1}}=x_{1}^{2}-y_{1}^{2}\].     (ii) Parametric form : The equation of the normal at \[\left( ct,\frac{c}{t} \right)\] to the hyperbola \[xy={{c}^{2}}\] is \[x{{t}^{3}}-yt-c{{t}^{4}}+c=0\].     On replacing \[{{x}_{1}}\] by \[ct\] and \[{{y}_{1}}\] by \[c/t\] in the equation.     We obtain \[x{{x}_{1}}-y{{y}_{1}}=x_{1}^{2}-y_{1}^{2},\]      \[xct-\frac{yc}{t}={{c}^{2}}{{t}^{2}}-\frac{{{c}^{2}}}{{{t}^{2}}}\Rightarrow x{{t}^{3}}-yt-c{{t}^{4}}+c=0\].     This equation is a fourth degree in \[t\]. So, in general four normals can be drawn from a point to the hyperbola \[xy={{c}^{2}}\], and point of intersection of normals at \[{{t}_{1}}\] and \[{{t}_{2}}\] is     \[\left( \frac{c\,\{{{t}_{1}}{{t}_{2}}(t_{1}^{2}+{{t}_{1}}{{t}_{2}}+t_{2}^{2})-1\}}{{{t}_{1}}{{t}_{2}}({{t}_{1}}+{{t}_{2}})},\,\,\frac{c\,\{t_{1}^{3}t_{2}^{3}+(t_{1}^{2}+{{t}_{1}}{{t}_{2}}+{{t}_{2}})\}}{{{t}_{1}}{{t}_{2}}({{t}_{1}}+{{t}_{2}})} \right)\].

Let \[\varphi (x)\] be the primitive or anti-derivative of a function \[f(x)\] defined on \[[a,\,\,b]\] i.e., \[\frac{d}{dx}[\varphi (x)]=f(x)\]. Then the definite integral of  \[f(x)\] over \[[a,\,\,b]\] is denoted by \[\int_{a}^{b}{f(x)dx}\] and is defined as \[[\varphi (b)-\varphi (a)]\] i.e., \[\int_{a}^{b}{f(x)dx=\varphi (b)-\varphi (a)}\]. This is also called Newton Leibnitz formula.   The numbers \[a\] and \[b\] are called the limits of integration, \['a'\] is called the lower limit and \['b'\] the upper limit. The interval \[[a,\,\,b]\]is called the interval of integration. The interval \[[a,\,\,b]\] is also known as range of integration. Every definite integral has a unique value.

When the variable in a definite integral is changed, the substitutions in terms of new variable should be effected at three places.     (i) In the integrand (ii)In the differential i.e., \[dx\] (iii) In the limits     For example, if we put \[\varphi (x)=t\] in the integral \[\int_{a}^{b}{f\{\varphi (x)\}\varphi '(x)dx}\], then \[\int_{a}^{b}{f\{\varphi (x)\}\varphi '(x)dx=\int_{\varphi (a)}^{\varphi (b)}{f(t)\,dt}}\].

(1) \[\int_{a}^{b}{f(x)dx}=\int_{a}^{b}{f(t)\,dt}\] i.e., The value of a definite integral remains unchanged if its variable is replaced by any other symbol.     (2) \[\int_{a}^{b}{f(x)dx=-\int_{b}^{a}{f(x)dx}}\] i.e., by the interchange in the limits of definite integral, the sign of the integral is changed.     (3) \[\int_{a}^{b}{f(x)dx=\int_{a}^{c}{f(x)dx}+\int_{c}^{b}{f(x)dx}}\],   (where \[a<c<b\])      or \[\int_{a}^{b}{f(x)dx}=\int_{a}^{{{c}_{1}}}{f(x)dx}+\int_{{{c}_{1}}}^{{{c}_{2}}}{f(x)dx+.....+\int_{{{c}_{n}}}^{b}{f(x)dx;}}\] (where \[a<{{c}_{1}}<{{c}_{2}}<........{{c}_{n}}<b\])     Generally this property is used when the integrand has two or more rules in the integration interval.     This is useful when \[f\,(x)\] is not continuous in \[[a,\,\,b]\] because we can break up the integral into several integrals at the points of discontinuity so that the function is continuous in the sub-intervals.     (4) \[\int_{0}^{a}{f(x)dx=\int_{0}^{a}{f(a-x)dx}}\] : This property can be used only when lower limit is zero. It is generally used for those complicated integrals whose denominators are unchanged when \[x\] is replaced by \[(a-x)\].     Following integrals can be obtained with the help of above property.     (i) \[\int_{0}^{\pi /2}{\frac{{{\sin }^{n}}x}{{{\sin }^{n}}x+{{\cos }^{n}}x}}\,dx=\int_{0}^{\pi /2}{\frac{{{\cos }^{n}}x}{{{\cos }^{n}}x+{{\sin }^{n}}x}dx=\frac{\pi }{4}}\]         (ii) \[\int_{0}^{\pi /2}{\frac{{{\tan }^{n}}x}{1+{{\tan }^{n}}x}dx=\int_{0}^{\pi /2}{\frac{{{\cot }^{n}}x}{1+{{\cot }^{n}}x}dx=\frac{\pi }{4}}}\]     (iii) \[\int_{0}^{\pi /2}{\frac{1}{1+{{\tan }^{n}}x}dx=\int_{0}^{\pi /2}{\frac{1}{1+{{\cot }^{n}}x}}dx=\frac{\pi }{4}}\]         (iv) \[\int_{0}^{\pi /2}{\frac{{{\sec }^{n\,}}x}{{{\sec }^{n}}\,x+\text{cose}{{\text{c}}^{n}}x}\,dx=}\int_{0}^{\pi /2}{\,\,\,\,}\frac{\text{cose}{{\text{c}}^{n\,}}x}{\text{cose}{{\text{c}}^{n}}\,x+{{\sec }^{n}}x}\,dx=\frac{\pi }{4}\]     (v) \[\int_{0}^{\pi /2}{\,\,\,f(\sin 2x)\sin xdx=}\int_{0}^{\pi /2}{\,\,\,f(\sin 2x)\cos xdx}\]     (vi) \[\int_{0}^{\pi /2}{f(\sin x)dx=\int_{0}^{\pi /2}{\,\,\,\,f(\cos x)dx}}\]              (vii) \[\int_{0}^{\pi /4}{\log (1+\tan x)dx=\frac{\pi }{8}\log 2}\]             (viii) \[\int_{0}^{\pi /2}{\,\,\,\,\,\log \sin xdx}=\int_{0}^{\pi /2}{\,\,\,\,\,\log \cos xdx}=\frac{-\pi }{2}\log 2=\frac{\pi }{2}\log \frac{1}{2}\]     (ix) \[\int_{0}^{\pi /2}{{}}\frac{a\,\sin \,x+b\,\cos \,x}{\sin \,x+\cos \,x}\,dx=\int_{0}^{\pi /2}{{}}\frac{a\,\sec \,x+b\,\text{cosec}\,x}{\sec \,x+\text{cosec}\,x}\,dx\]\[\int_{0}^{\pi /2}{{}}\frac{a\,\sin \,x+b\,\cos \,x}{\sin \,x+\cos \,x}\,dx=\int_{0}^{\pi /2}{{}}\frac{a\,\sec \,x+b\,\cos ec\,x}{\sec \,x+\cos ec\,x}\,dx=\int_{0}^{\pi /2}{{}}\frac{a\,\tan \,x+b\,\cot \,x}{\tan \,x+\cot \,x}\,dx=\frac{\pi }{4}(a+b)\]     (5) \[\int_{-a}^{a}{{}}f(x)\,dx=\int_{0}^{a}{\text{ }\!\![\!\!\text{ }f(x)\,+f(-x)\text{ }\!\!]\!\!\text{ }}\text{ }dx\].     In special case :     \[\int_{-a}^{a}{f(x)\,dx}=\left\{ \begin{array}{*{35}{l}} 2\int_{0}^{a}{f(x)\,dx},\,\,\text{if}\,f(x)\,\,\text{is}\,\,\text{even function or }f(-x)=f(x)\text{ }  \\ \,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\text{if}\,\,f(x)\,\,\text{is odd function or }f(-x)=-f(x)  \\\end{array} \right.\]       This property is generally used when integrand is either even or odd function of \[x\].     (6) \[\int_{0}^{2a}{\,\,f(x)dx}=\int_{0}^{a}{f(x)dx+\int_{0}^{a}{{}}\text{ }f(2a-x)\,dx}\]     In particular, \[\int_{0}^{2a}{{}}\,f(x)\,dx\,=\,\left\{ \begin{array}{*{35}{l}} 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\text{if}\,\,\,\,f(2a- x)=-f(x)  \\ 2\int_{0}^{a}{f(x)\,dx}\,,\,\,\,\text{if}\,\,\,\,\,f(2a-x)=f(x)  \\ \end{array} \right.\]     It is generally used to make half the upper limit.     (7) \[\int_{a}^{b}{f(x)\,}dx=\int_{a}^{b}{f(a+b-x)dx}\].     (8) \[\int_{0}^{a}{\,\,x\,f(x)dx}=\frac{1}{2}a\int_{0}^{a}{\,\,f(x)dx}\], if \[f(a-x)=f(x)\].     (9) If \[f(x)\] is a periodic function with period T, then \[\int_{0}^{nT}{f(x)dx=n\int_{0}^{T}{\,\,f(x)dx}}\]     Deduction : If \[f(x)\] is a periodic function with period T, then     \[\int_{a}^{a+nT}{{}}f(x)\,dx=n\,\int_{0}^{T}{{}}f(x)\,dx\] , where \[n\,\in \,I\]     (a) If \[a=0,\] \[\int_{0}^{nT}{{}}f(x)\,dx=n\int_{0}^{T}{{}}f(x)\,dx,\] where \[n\in I\]     (b) If \[n=1,\] \[\int_{0}^{a+T}{{}}f(x)\,dx=\int_{0}^{T}{{}}f(x)\,dx\].     (10) \[\int_{mT}^{nT}{{}}f(x)\,dx=(n-m)\,\,\int_{0}^{T}{{}}f(x)\,dx,\] where \[n,\] \[m\in I\].     (11) \[\int_{a+nT}^{b+nT}{{}}f(x)\,dx=\int_{a}^{b}{{}}f(x)\,dx,\] where \[n\in \,I\].     (12) \[\int_{0}^{2k}{(x-[x])\,dx=k,}\] where \[k\] an integer, since \[x-[x]\] is a periodic function with period 1.     (13) If \[f(x)\] is a periodic function with period T, then \[\int_{a}^{a+T}{{}}f(x)\] is independent of a.     (14) \[\int_{a}^{b}{{}}f(x)\,dx=(b-a)\,\int_{0}^{1}{{}}f((b-a)\,x+a)\,dx\].

We know that \[\int_{a}^{b}{f(x)dx}=\underset{n\to \infty }{\mathop{\lim }}\,h\sum\limits_{r=1}^{n}{f(a+rh)}\], where \[nh=b-a\]     Now, put  \[a=0,\] \[b=1,\] \[\therefore \] \[nh=1\] or \[h=\frac{1}{n}\].     Hence \[\int_{0}^{1}{f(x)\,dx=\underset{n\to \infty }{\mathop{\lim }}\,}\frac{1}{n}\sum\limits_{{}}^{{}}{f\left( \frac{r}{n} \right)}\].     Express the given series in the form \[\sum{\frac{1}{n}f\left( \frac{r}{h} \right)}\].     Replace \[\frac{r}{n}\]by \[x,\,\,\frac{1}{n}\] by \[dx\] and the limit of the sum is \[\int_{0}^{1}{{}}f(x)\,dx\].

\[\int_{0}^{\infty }{{{x}^{n-1}}}{{e}^{-x}}dx\], \[n>0\] is called Gamma function and denoted by \[\Gamma n\]. If \[m\] and \[n\] are non-negative integers, then     \[\int_{0}^{\pi /2}{{{\sin }^{m}}x{{\cos }^{n}}xdx}=\frac{\Gamma \left( \frac{m+1}{2} \right)\,\Gamma \left( \frac{n+1}{2} \right)}{2\Gamma \left( \frac{m+n+2}{2} \right)}\]     where \[\Gamma (n)\] is called gamma function which satisfy the following properties \[\Gamma (n+1)=n\Gamma (n)=n!\]i.e.,\[\Gamma \,(1)=1\], \[\Gamma (1/2)=\sqrt{\pi }\]     In place of gamma function, we can also use the following formula \[\int_{0}^{\pi /2}{{{\sin }^{m}}x{{\cos }^{n}}xdx}\]     =\[\frac{(m-1)(m-3).....(2\text{ or }1)(n-1)(n-3).....(2\text{ or 1)}}{(m+n)(m+n-2)....(2\text{ or }1)}\]     It is important to note that we multiply by \[(\pi /2);\] when both \[m\] and \[n\] are even.