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(1) Point form : The equation of the tangent to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at \[({{x}_{1}},\,{{y}_{1}})\] is \[\frac{x{{x}_{1}}}{{{a}^{2}}}-\frac{y{{y}_{1}}}{{{b}^{2}}}=1\].     (2) Parametric form : The equation of tangent to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at \[(a\sec \varphi ,\,b\tan \varphi )\] is \[\frac{x}{a}\sec \varphi -\frac{y}{b}\tan \varphi =1\].     (3) Slope form : The equations of tangents of slope m to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are \[y=mx\pm \sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}\] and the co-ordinates of points of contacts are \[\left( \pm \frac{{{a}^{2}}m}{\sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}},\,\pm \frac{{{b}^{2}}}{\sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}}\, \right)\].

If \[P({{x}_{1}},\,{{y}_{1}})\] be any point outside the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] then a pair of tangents PQ, PR can be drawn to it from P.     The equation of pair of tangents PQ and PR is \[S{{S}_{1}}={{T}^{2}}\]        where,\[S=\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}-1\],\[{{S}_{1}}=\frac{x_{1}^{2}}{{{a}^{2}}}-\frac{y_{1}^{2}}{{{b}^{2}}}-1,\,T=\frac{x{{x}_{1}}}{{{a}^{2}}}-\frac{y{{y}_{1}}}{{{b}^{2}}}-1\]     Director circle : The director circle is the locus of points from which perpendicular tangents are drawn to the given hyperbola. The equation of the director circle of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}-{{b}^{2}}\].

(1) Point form : The equation of normal to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at \[({{x}_{1}},\,{{y}_{1}})\] is \[\frac{{{a}^{2}}x}{{{x}_{1}}}+\frac{{{b}^{2}}y}{{{y}_{1}}}={{a}^{2}}+{{b}^{2}}\].     (2) Parametric form: The equation of normal at \[(a\sec \theta ,\,b\tan \theta )\] to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[ax\cos \theta +by\cot \theta ={{a}^{2}}+{{b}^{2}}\]     (3) Slope form: The equation of the normal to the hyperbola  \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] in terms of the slope m of the normal is \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\].     (4) Condition for normality : If \[y=mx+c\] is the normal of \[S=0\], then \[c=\mp \frac{m({{a}^{2}}+{{b}^{2}})}{\sqrt{{{a}^{2}}-{{m}^{2}}{{b}^{2}}}}\] or \[{{c}^{2}}=\frac{{{m}^{2}}{{({{a}^{2}}+{{b}^{2}})}^{2}}}{({{a}^{2}}-{{m}^{2}}{{b}^{2}})}\], which is condition of normality.     (5) Points of contact : Co-ordinates of points of contact are \[\left( \pm \frac{{{a}^{2}}}{\sqrt{{{a}^{2}}-{{b}^{2}}{{m}^{2}}}},\,\mp \frac{m{{b}^{2}}}{\sqrt{{{a}^{2}}-{{b}^{2}}{{m}^{2}}}} \right)\].

Let PQ and PR be tangents to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] drawn from any external point \[P\,({{x}_{1}},\,{{y}_{1}})\].     Then equation of chord of contact QR is \[\frac{x{{x}_{1}}}{{{a}^{2}}}-\frac{y{{y}_{1}}}{{{b}^{2}}}=1\] or \[T=0\],          

Equation of the chord of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\], bisected at the given point \[({{x}_{1}},\,{{y}_{1}})\] is \[\frac{x{{x}_{1}}}{{{a}^{2}}}-\frac{y{{y}_{1}}}{{{b}^{2}}}-1=\frac{x_{1}^{2}}{{{a}^{2}}}-\frac{y_{1}^{2}}{{{b}^{2}}}-1\] i.e., \[T={{S}_{1}}\].      

The equation of the chord joining the points \[P(a\sec {{\varphi }_{1}},\,\,b\tan {{\varphi }_{1}})\] and \[Q(a\,\sec {{\varphi }_{2}},\,b\tan {{\varphi }_{2}})\] is \[\frac{x}{a}\cos \left( \frac{{{\varphi }_{1}}-{{\varphi }_{2}}}{2} \right)-\frac{y}{b}\sin \left( \frac{{{\varphi }_{1}}+{{\varphi }_{2}}}{2} \right)=\cos \left( \frac{{{\varphi }_{1}}+{{\varphi }_{2}}}{2} \right)\].  

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     \[BN=CB-CN=\frac{({{a}^{2}}+{{b}^{2}})}{{{a}^{2}}}{{x}_{1}}-{{x}_{1}}\]     =\[\frac{{{b}^{2}}}{{{a}^{2}}}{{x}_{1}}=({{e}^{2}}-1){{x}_{1}}\].

Let the circle be \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] and line mirror \[lx+my+n=0\]. In this condition, radius of circle remains unchanged but centre changes. Let the centre of image circle be \[({{x}_{1}},\,\,{{y}_{1}})\]. Slope of \[{{C}_{1}}{{C}_{2}}\times \] (slope of \[lx+my+n=0)=-1\]            …..(i)     and mid point of \[{{C}_{1}}(-g,\,-f)\] and \[{{C}_{2}}({{x}_{1}},{{y}_{1}})\] lies on \[lx+my+n=0\]     i.e.,  \[l\,\left( \frac{{{x}_{1}}-g}{2} \right)\,+m\,\left( \frac{{{y}_{1}}-f}{2} \right)\,+\,n=0\]                   …..(ii)     Solving (i) and (ii), we get \[({{x}_{1}},\,\,{{y}_{1}})\]     \[\therefore \] Required image circle is \[{{(x-{{x}_{1}})}^{2}}+{{(y-{{y}_{1}})}^{2}}={{r}^{2}}\],  where \[r=\sqrt{({{g}^{2}}+{{f}^{2}}-c)}\]