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Let \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] be the hyperbola, then equation of the auxiliary circle is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\].     Let \[\angle QCN=\varphi \]. Here P and Q are the corresponding points on the hyperbola and the auxiliary circle \[(0\le \varphi <2\pi )\].  

The equations \[x=a\sec \varphi \] and \[y=b\tan \varphi \] are known as the parametric equations of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]. This \[(a\sec \varphi ,\,b\tan \varphi )\] lies on the hyperbola for all values of \[\varphi \].  

Let the hyperbola be \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\].     Then \[P({{x}_{1}},\,{{y}_{1}})\] will lie inside, on or outside the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] according as \[\frac{x_{1}^{2}}{{{a}^{2}}}-\frac{y_{1}^{2}}{{{b}^{2}}}-1\] is positive, zero or negative.  

The straight line \[y=mx+c\] will cut the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] in two points may be real, coincident or imaginary according as \[{{c}^{2}}>,\,=,\,<{{a}^{2}}{{m}^{2}}-{{b}^{2}}\].     Condition of tangency : If straight line \[y=mx+c\] touches the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then \[{{c}^{2}}={{a}^{2}}{{m}^{2}}-{{b}^{2}}\].

(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 \[\frac{x}{a}\cos \left( \frac{{{\varphi }_{1}}-{{\varphi }_{2}}}{2} \right)-\frac{y}{b}\sin \left( \frac{{{\varphi }_{1}}+{{\varphi }_{2}}}{2} \right)=\cos \left( \frac{{{\varphi }_{1}}+{{\varphi }_{2}}}{2} \right)\].