Equation of Tangent – Ellipse
Equation of tangent to the ellipse at point (x₁, y₁).
\(\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\),
Differentiation with respect to x
\(\frac{2x}{{{a}^{2}}}+\frac{2y}{{{b}^{2}}}.\frac{dy}{dx}=0\),
\(\frac{dy}{dx}=-\frac{\frac{2x}{{{a}^{2}}}}{\frac{2y}{{{b}^{2}}}}\),
\(\frac{dy}{dx}=-\frac{{{b}^{2}}x}{{{a}^{2}}y}\),
dy/dx at point (x₁, y₁)
\(\frac{dy}{dx}=-\frac{{{b}^{2}}{{x}_{1}}}{{{a}^{2}}{{y}_{1}}}\),
Hence the equation of the tangent is
\(y-{{y}_{1}}=-\frac{{{b}^{2}}{{x}_{1}}}{{{a}^{2}}{{y}_{1}}}.(x-{{x}_{1}})\),
\(\frac{x{{x}_{1}}}{{{a}^{2}}}+\frac{y{{y}_{1}}}{{{b}^{2}}}=\frac{{{x}_{1}}^{2}}{{{a}^{2}}}+\frac{{{y}_{1}}^{2}}{{{b}^{2}}}\),
But (x₁, y₁) lies on the ellipse
∴ \(\frac{{{x}_{1}}^{2}}{{{a}^{2}}}+\frac{{{y}_{1}}^{2}}{{{b}^{2}}}=1\),
\(\frac{{{x}_{1}}^{2}}{{{a}^{2}}}+\frac{{{y}_{1}}^{2}}{{{b}^{2}}}-1=0\),
(or)
T = 0
Where, \(T\text{ }=\frac{x{{x}_{1}}}{{{a}^{2}}}+\frac{y{{y}_{1}}}{{{b}^{2}}}\).