The Locus of the Point of Intersection of the Perpendicular Tangents to the Parabola x^2 = 4ay

Introduction:
A parabola is a u-shaped curve that can be defined by a quadratic equation. In this case, the equation of the parabola is x^2 = 4ay, where "a" is a constant. We are interested in finding the locus of the point of intersection of the perpendicular tangents to this parabola.

Understanding Tangents to a Parabola:
A tangent to a curve is a straight line that touches the curve at a single point. For a parabola, there can be multiple tangents at different points along the curve. However, we are specifically interested in the perpendicular tangents, which are perpendicular to the tangent line at the point of contact.

Finding the Tangent to the Parabola:
To find the tangent to the parabola, we need to differentiate the equation of the parabola with respect to x. Differentiating x^2 = 4ay with respect to x, we get:

2x = 4a(dy/dx)

Simplifying, we find:

dy/dx = x/2a

This is the slope of the tangent line at any point (x,y) on the parabola.

Finding the Equation of the Perpendicular Tangent:
To find the equation of the perpendicular tangent, we need to find the negative reciprocal of the slope of the tangent line. The negative reciprocal of x/2a is -2a/x.

Using the point-slope form of a line (y - y1) = m(x - x1), we can write the equation of the perpendicular tangent as:

y - y1 = (-2a/x)(x - x1)

Where (x1, y1) is any point on the parabola.

Finding the Intersection Point of Perpendicular Tangents:
To find the point of intersection of the perpendicular tangents, we need to find the values of x and y that satisfy both tangent equations. Let's consider two different points on the parabola, (x1, y1) and (x2, y2).

For the first tangent equation, we have:

y - y1 = (-2a/x1)(x - x1)

And for the second tangent equation, we have:

y - y2 = (-2a/x2)(x - x2)

Setting the two equations equal to each other, we get:

(-2a/x1)(x - x1) + y1 = (-2a/x2)(x - x2) + y2

Expanding and simplifying, we find:

(-2a/x1)x + (2a/x2)x = (-2a/x2)x1 + (-2a/x1)x2 + y2 - y1

Which further simplifies to:

(-2a/x1)x + (2a/x2)x = (-2a/x2)x1 + (-2a/x1)x2

Dividing both sides by a, we get:

(-2/x1)x + (2/x2)x = (-2/x2)x1 + (-2/x1)x2

Simplifying further, we find:

(-2/x1)x + (2/x2)x = (-2/x2)x1 + (-2/x1)x2

Finally, we