Normal at Eccentric Angle θ

Normal at (a cosθ, b sinθ). Slope of normal = (a sinθ)/(b cosθ) = (a/b) tanθ. Passes through the centre only when θ = 0°, 90°, 180°, 270° (the vertices).

Substitute $x_1 = a\cos\theta$, $y_1 = b\sin\theta$ into the point-form normal $a^2x/x_1 - b^2y/y_1 = a^2-b^2$:

$$\frac{a^2x}{a\cos\theta} - \frac{b^2y}{b\sin\theta} = a^2 - b^2$$ $$ax\sec\theta - by\csc\theta = a^2 - b^2$$

Slope: Rewriting in $y = mx + c$ form:

$$by\csc\theta = ax\sec\theta - (a^2-b^2)$$ $$y = \frac{a\sin\theta}{b\cos\theta}x - \frac{(a^2-b^2)\sin\theta}{b}$$

Slope of normal $= \dfrac{a\sin\theta}{b\cos\theta} = \dfrac{a}{b}\tan\theta$.

Foot of normal on major axis: Set $y = 0$:

$$ax\sec\theta = a^2-b^2 = c^2 \implies x = c^2\cos\theta/a = ae^2\cos\theta$$

Foot is $(ae^2\cos\theta, 0)$. This is always inside the ellipse on the major axis.

Four co-normal points: Up to four normals to an ellipse can pass through a given point (inside the evolute). This contrasts with the parabola (three) and the hyperbola (also up to four).