Point of Intersection of Normals

In the study of conic sections, particularly parabolas, the concept of normals is crucial. A normal to a curve at a given point is a line perpendicular to the tangent at that point. When dealing with parabolas, normals drawn at different points can intersect, and the point of intersection of these normals is a topic of interest in coordinate geometry.

Understanding Normals

Before diving into the intersection of normals, let's understand what a normal is in the context of a parabola.

Definition of a Normal

For a curve defined by the equation $y = f(x)$, a normal at a point $(x_0, y_0)$ on the curve is a line that is perpendicular to the tangent line at that point. The slope of the normal is the negative reciprocal of the slope of the tangent.

If the slope of the tangent is $m_t$, then the slope of the normal $m_n$ is given by:

$$ m_n = -\frac{1}{m_t} $$

For a parabola defined by the equation $y^2 = 4ax$, the slope of the tangent at a point $(x_1, y_1)$ is given by $\frac{y_1}{2a}$, and thus the slope of the normal at that point is $-\frac{2a}{y_1}$.

Equation of a Normal

The equation of a normal to the parabola $y^2 = 4ax$ at the point $(x_1, y_1)$ can be written as:

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

Intersection of Normals

When more than one normal is drawn to a parabola, these lines may intersect at a common point. The point of intersection of normals is not arbitrary and can be determined algebraically.

Finding the Point of Intersection

To find the point of intersection of two or more normals, we need to solve the system of equations representing those normals simultaneously.

Example

Consider the parabola $y^2 = 4ax$. Let's find the point of intersection of the normals at points $P(t_1)$ and $Q(t_2)$ on the parabola.

The equations of the normals at $P$ and $Q$ are:

$$ y + 2at_1x = 2at_1^2 - a(t_1^4) $$

$$ y + 2at_2x = 2at_2^2 - a(t_2^4) $$

To find the point of intersection, we solve these two equations simultaneously.

Table of Differences and Important Points

Feature	Tangent	Normal
Definition	A line that touches the curve at a single point without crossing it.	A line perpendicular to the tangent at the point of contact.
Slope (for parabola $y^2 = 4ax$)	$m_t = \frac{y_1}{2a}$	$m_n = -\frac{2a}{y_1}$
Equation (at point $(x_1, y_1)$)	$yy_1 = 2a(x + x_1)$	$y - y_1 = -\frac{2a}{y_1}(x - x_1)$
Intersection	Tangents from an external point intersect at this point.	Normals can intersect at a point different from the points of tangency.

Formulas

The general formula for the equation of a normal to a parabola $y^2 = 4ax$ at a point $(at^2, 2at)$ is:

$$ y = -tx + 2at + at^3 $$

To find the intersection of two normals, we solve the system:

$$ \begin{align*} y &= -t_1x + 2at_1 + at_1^3 \ y &= -t_2x + 2at_2 + at_2^3 \end{align*} $$

Examples

Example 1: Intersection of Two Normals

Given the parabola $y^2 = 4x$, find the point of intersection of the normals at points where $t_1 = 1$ and $t_2 = -1$.

Solution:

Equations of the normals are:

$$ y = -x + 2 + 1 $$

$$ y = x - 2 + 1 $$

Solving these simultaneously, we get $y = 3$ and $x = 0$. Thus, the point of intersection is $(0, 3)$.

Example 2: Intersection of Three Normals

Given the parabola $y^2 = 12x$, find the point of intersection of the normals at points where $t_1 = 2$, $t_2 = -2$, and $t_3 = 3$.

Solution:

Equations of the normals are:

$$ y = -2x + 12 + 16 $$

$$ y = 2x - 12 + 16 $$

$$ y = -3x + 18 + 54 $$

Solving the first two equations, we get $y = 28$ and $x = 0$. Checking with the third equation, we see that it also passes through $(0, 28)$. Thus, the point of intersection is $(0, 28)$.

In conclusion, the point of intersection of normals to a parabola is a significant concept in coordinate geometry, especially when dealing with problems involving multiple normals. The intersection point can be found by solving the equations of the normals simultaneously.