JEE Main & Advanced Mathematics Conic Sections Equations of Tangent in Different Forms

(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 \[\left( \pm \frac{{{a}^{2}}m}{\sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}},\,\pm \frac{{{b}^{2}}}{\sqrt{{{a}^{2}}{{m}^{2}}-{{b}^{2}}}}\, \right)\].