# Equation of Tangent – Ellipse

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