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

(1) Point form: The equation of the normal at \[({{x}_{1}},{{y}_{1}})\]to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]is \[\frac{{{a}^{2}}x}{{{x}_{1}}}-\frac{{{b}^{2}}y}{{{y}_{1}}}={{a}^{2}}-{{b}^{2}}\].

(2) Parametric form: The equation of the normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at \[(a\cos \varphi ,b\sin \varphi )\] is \[ax\sec \varphi -by\,\text{cos}\text{ec}\varphi =\] \[{{a}^{2}}-{{b}^{2}}\].

(3) Slope form: If m is the slope of the normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then the equation of normal is \[y=mx\pm \frac{m({{a}^{2}}-{{b}^{2}})}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}}\].

The co-ordinates of the point of contact are  \[\left( \frac{\pm {{a}^{2}}}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}},\frac{\pm m{{b}^{2}}}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}} \right)\] .