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

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