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

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

(2) Slope form: If the line \[y=mx+c\]touches the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then \[{{c}^{2}}={{a}^{2}}{{m}^{2}}+{{b}^{2}}\]. Hence, the straight line \[y=mx\pm \sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\]always represents the tangents to the ellipse.

Points of contact: Line \[y=mx\pm \sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\] touches the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at  \[\left( \frac{\pm {{a}^{2}}m}{\sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}},\frac{\mp {{b}^{2}}}{\sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}} \right)\].

(3) Parametric form: The equation of tangent at any point \[(a\cos \varphi ,b\sin \varphi )\] is \[\frac{x}{a}\cos \varphi +\frac{y}{b}\sin \varphi =1\].