Condition for Three Normals

In the study of conic sections, particularly parabolas, normals play a significant role. A normal to a curve at a given point is a line perpendicular to the tangent at that point. For a parabola, there can be up to three distinct normals drawn from a point outside the parabola. Understanding the conditions for these three normals is important for various applications in geometry and calculus.

Parabola and Its Equation

A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). The standard equation of a parabola with its vertex at the origin and axis parallel to the y-axis is:

$$ y^2 = 4ax $$

where (a) is the distance from the vertex to the focus (and also to the directrix).

Equation of the Normal

The equation of the normal to the parabola (y^2 = 4ax) at the point ((at^2, 2at)) is given by:

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

This equation can be rewritten as:

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

Condition for Three Normals

For a given point (P(x_1, y_1)) outside the parabola, there can be three, one, or no normals drawn to the parabola. The condition for the existence of three distinct normals from point (P) to the parabola (y^2 = 4ax) is that the cubic equation in (t):

$$ y_1 + tx_1 = 2at + at^3 $$

must have three real and distinct roots. This is because each real root (t) corresponds to a point on the parabola where a normal can be drawn.

Table of Conditions

Number of Normals	Condition on Roots of the Equation	Geometric Interpretation
Three	Three real and distinct roots	Point (P) is outside and not too close to the parabola
One	One real root and two complex roots	Point (P) lies on the parabola or is very close to it
None	All roots are complex	Point (P) is inside the parabola

Examples

Example 1: Three Distinct Normals

Consider the point (P(8, 6)) and the parabola (y^2 = 4x). The equation for the normals is:

$$ 6 + 8t = 2t + t^3 $$

Simplifying, we get:

$$ t^3 - 6t + 6 = 0 $$

This cubic equation must be solved to find the values of (t). If it has three distinct real roots, then there are three normals from (P) to the parabola.

Example 2: One Normal

Consider the point (P(2, 4)) on the parabola (y^2 = 4x). The equation for the normals is:

$$ 4 + 2t = 2t + t^3 $$

Simplifying, we get:

$$ t^3 = 0 $$

This equation has one real root, (t = 0), and thus there is only one normal to the parabola at the point (P).

Conclusion

The condition for three normals from a point to a parabola is an important concept in the study of conic sections. It is essential for solving problems related to the geometry of parabolas and has applications in various fields such as physics, engineering, and computer graphics. Understanding the conditions and being able to apply them to solve equations is a key skill for students and professionals working with these mathematical structures.