Locus of the point of intersection of tangents drawn at the extremities of normal chords to the hyperbola xy = c^2a(x^2 - y^2)^2 is given by option A.

To understand why option A is the correct answer, let's break down the problem step by step.

1. Equation of the Hyperbola:
The given equation is xy = c^2a(x^2 - y^2)^2. This is the equation of a hyperbola with its center at the origin (0, 0).

2. Normal Chords:
A chord is a line segment that joins two points on a curve. In this case, we are interested in normal chords, which are perpendicular to the tangent lines drawn at their extremities.

3. Tangents to the Hyperbola:
To find the tangents to the hyperbola, we need to differentiate the equation of the hyperbola with respect to x and y.

Differentiating with respect to x, we get:
y + xy' = c^2a(2x(x^2 - y^2) + (x^2 - y^2)^2)
Simplifying this equation, we get:
y' = c^2a(2x(x^2 - y^2) + (x^2 - y^2)^2)/(1 + xy)

Similarly, differentiating with respect to y, we get:
x + xy' = c^2a(-2y(x^2 - y^2) + (x^2 - y^2)^2)
Simplifying this equation, we get:
x' = c^2a(-2y(x^2 - y^2) + (x^2 - y^2)^2)/(1 + xy)

These equations represent the slopes of the tangents to the hyperbola at any point (x, y).

4. Intersection of Tangents:
To find the point of intersection of tangents, we equate the slopes obtained from the above equations. This gives us a quadratic equation in terms of x and y.

Simplifying the equation, we get:
2x(x^2 - y^2) + (x^2 - y^2)^2 = -2y(x^2 - y^2) + (x^2 - y^2)^2

Canceling out the common factors, we get:
2x = -2y

Dividing both sides by 2, we get:
x = -y

This equation represents the locus of the point of intersection of tangents drawn at the extremities of normal chords to the hyperbola.

5. Comparing with the Given Options:
Option A states (x^2 - y^2) = 4cxy = 0. If we rearrange this equation, we get:
x^2 - y^2 = 4cxy

Comparing this with the equation x = -y obtained earlier, we can see that option A is the correct answer.

Therefore, the locus of the point of intersection of tangents drawn at the extremities of normal chords to the hyperbola xy = c^2a(x^2 - y^2)^2 is given by option A.