Pair of Tangents from an External Point

Combined equation of tangents from (x₁, y₁) to x²/a² + y²/b² = 1, where S = x²/a² + y²/b² − 1, S₁ = x₁²/a² + y₁²/b² − 1, T = xx₁/a² + yy₁/b² − 1.

For ellipse $x^2/a^2 + y^2/b^2 = 1$, define:

$$S \equiv \frac{x^2}{a^2} + \frac{y^2}{b^2} - 1, \quad S_1 \equiv \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} - 1, \quad T \equiv \frac{xx_1}{a^2} + \frac{yy_1}{b^2} - 1$$

The combined equation of the two tangents from external point $(x_1, y_1)$ is:

$$SS_1 = T^2$$

Derivation sketch: Any point $Q(h, k)$ on either tangent from $P(x_1, y_1)$ satisfies the condition that line $PQ$ is tangent to the ellipse. Substituting the parametric form of $PQ$ into $S = 0$ and setting discriminant to zero, after simplification, yields $S(Q)\cdot S_1 = T(Q)^2$.

This result is universal: The form $SS_1 = T^2$ holds for circles, parabolas, ellipses, and hyperbolas. Memorising the pattern rather than rederiving each time is the practical approach.

Angle between the tangents from $(x_1, y_1)$:

$$\tan\theta = \frac{2\sqrt{S_1(a^2+b^2) - a^2b^2S_1^2/S_1}}{(\text{sum of squared terms})}$$

In practice, the angle is found by extracting the pair equation and applying the standard formula for angle between a pair of lines.