Equation of Tangent – Ellipse

Equation of Tangent – Ellipse

Equation of tangent to the ellipse at point (x₁, y₁).

Equation of Tangent - Ellipse

\(\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\),

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,  \(T\text{ }=\frac{x{{x}_{1}}}{{{a}^{2}}}+\frac{y{{y}_{1}}}{{{b}^{2}}}\).