Normal at a Point on the Parabola

Normal to y² = 4ax at (x₁, y₁). Slope of normal = −y₁/2a (negative reciprocal of the tangent slope 2a/y₁).

At $P(x_1, y_1)$ on $y^2 = 4ax$, the tangent slope is $2a/y_1$. The normal, being perpendicular to the tangent, has slope:

$$m_{\text{normal}} = -\frac{y_1}{2a}$$

Equation of the normal through $P(x_1, y_1)$:

$$y - y_1 = -\frac{y_1}{2a}(x - x_1)$$

Key difference from circles: The normal to a circle always passes through the centre. The normal to a parabola does not (in general) pass through the focus. This is because the parabola is not a "centre-symmetric" conic.

Where the normal meets the axis: Set $y = 0$:

$$-y_1 = -\frac{y_1}{2a}(x - x_1) \implies 2a = x - x_1 \implies x = x_1 + 2a$$

The foot of the normal on the axis is $(x_1 + 2a, 0)$. In parametric terms (with $x_1 = at^2$): foot is at $(at^2 + 2a, 0) = (a(t^2+2), 0)$.

Subnormal: The projection of the normal segment between the point and the axis onto the axis has length:

$$(x_1 + 2a) - x_1 = 2a = \text{constant}$$

The subnormal of a parabola is constant, equal to $2a$. This is a classical property.