# Ellipse – Tangent

## Ellipse – Tangent

Tangent to the Ellipse: Let S = 0 be an ellipse and P be a point on the ellipse.

Let Q be any other point on the ellipse. If the secant line PQ approaches to the same limiting position as Q moves along the curve and approaches to P from either side, then the limiting position is called a tangent line or tangent to the ellipse at P. the point P is called of contact of the tangent to the ellipse. If L = 0 is a tangent to the ellipse S = 0 at P, then we say that the line L = 0 touches the ellipse S = 0 at P.

Theorem: The equation of the tangent to the ellipse S = 0 at P (x₁, y₁) is S₁ = 0

Proof: P (x₁, y₁) is a point on the ellipse S = 0

S₁₁ = 0

Let Q (x₂, y₂) be a point on the ellipse S = 0

The equation of chord joining P, Q is S₁ + S₂ = S₁₂.

If Q approaches to P then the chord

PQ becomes the tangent at P is

∴ The equation of the tangent at P is \underset{Q\to P}{\mathop{\lim }}\,\{{{S}_{1}}+{{S}_{2}}={{S}_{12}}\}[\latex].

S₁ + S₁ = S₁₁

2S₁ = 0

S₁ = 0

Example: Find the equation of the tangent to the ellipse.x² + 8y² = 33 at (-1, 2)

Solution: Given,

x² + 8y² = 33 at (-1, 2)

The equation of the tangent to the ellipse S = 0 at P (x₁, y₁) is S₁ = 0

xx₁+ 8yy₁ = 33 at (-1, 2)

x₁ = -1, y₁ = 2

x (-1) + 8y (2) = 33

– x + 16y = 33

x – 16y + 33 = 0.