Determining the Locus of the Point of Intersection of Two Tangents to a Parabola Forming a 45 Degree Angle

Introduction

In this article, we will delve into the fascinating problem of determining the locus of the point of intersection of two tangents to a parabola that meet at a 45-degree angle. We will explore the mathematical derivation and findings in detail, providing insights into the geometric properties and algebraic expressions involved.

Understanding the Problem

Consider a parabola with the standard form equation:

y2 4ax

where a is a constant.

Equation of Tangents

The equation for a tangent to the parabola at a point (at2, 2at) is given by:

y t2x 2at

Here, t is a parameter corresponding to the point of tangency.

Two Tangents at Different Points

Let's consider two tangents at points corresponding to parameters t1 and t2:The first tangent: y t1x 2at1 The second tangent: y t2x 2at2

Intersection Point of Two Tangents

To find the intersection point of these two tangents, we set their equations equal to each other:

2at1 t1x 2at2 t2x

Rearranging, we get:

xt1 - xt2 2at2 - 2at1

Factoring out x and 2a, we obtain:

x(t1 - t2) 2a(t2 - t1)

Therefore, the x-coordinate of the intersection point is:

x -2at2 / (t1 - t2)

Now, substituting this back into one of the tangent equations to find y:

y t1(-2at2 / (t1 - t2)) 2at1

Simplifying, we get:

y -2at2 / (t1 - t2) 2at1

Angle Between Two Tangents

The angle θ between the two tangents can be found using the formula:

tan θ (t1 - t2) / (1 t1t2)

Given that the angle is 45 degrees, θ 45°, we have:

tan 45° 1 (t1 - t2) / (1 t1t2)

This leads to two cases:

(t1 - t2) 1 t1t2 (t1 - t2) -(1 t1t2)

Locus of the Intersection Point

To find the locus of the intersection point, we need to eliminate t1 and t2. We use the relationships s t1 t2 and p t1t2.

After some algebraic manipulation, we find that the locus of the intersection point satisfies the equation of a new parabola:

y2 2ax

Thus, the locus of the point of intersection of two tangents to the parabola y2 4ax that meet at a 45-degree angle is given by:

y2 2ax

Conclusion

The problem of finding the locus of the point of intersection of two tangents to a parabola forming a 45-degree angle has been explored through the provided derivation. The final result is a new parabolic equation, which illustrates the intricate relationship between geometric figures and algebraic expressions.