The Locus of Point of Intersection of Tangents at the Ends of Normal Chord of Hyperbola

The locus of the point of intersection of tangents at the ends of a normal chord of a hyperbola can be determined by considering the properties of the hyperbola and the concept of the normal chord.

Definition of a Hyperbola:
A hyperbola is a conic section defined as the set of all points in a plane, the difference of whose distances from two fixed points (called foci) is constant.

Equation of a Hyperbola:
The equation of a hyperbola with center (h, k), semi-major axis a, and semi-minor axis b can be written as:

[(x - h)² / a²] - [(y - k)² / b²] = 1 --(1)

Given Hyperbola:
For the given hyperbola with equation x² - y² = a², we can rewrite it in the standard form by dividing the equation by a²:

[(x - 0)² / a²] - [(y - 0)² / a²] = 1

From this equation, we can observe that h = 0, k = 0, and a = a.

Finding the Normal Chord:
A normal chord of a hyperbola is a line segment that passes through the focus and is perpendicular to the tangent at that point.

Equation of Tangent to Hyperbola:
The equation of the tangent to a hyperbola at any point (x₁, y₁) on the hyperbola can be determined by differentiating the equation of the hyperbola and substituting the coordinates of the point:

dy/dx = (2x₁) / (-2y₁) --(2)

The equation of the tangent can be written as:

y - y₁ = (dy/dx) * (x - x₁)

Substituting the value of dy/dx from equation (2), we get:

y - y₁ = (x₁ / y₁) * (x - x₁)

Simplifying this equation, we obtain:

y = [(x₁ / y₁) * x] - [(x₁ / y₁) * x₁] + y₁

y = (x₁ / y₁) * x - [(x₁² / y₁) - y₁]

Equation of Normal Chord:
The equation of the normal chord can be determined by finding the slope of the tangent at a particular point and taking the negative reciprocal of it. Let's assume the coordinates of the point on the hyperbola are (x₁, y₁).

The slope of the tangent at (x₁, y₁) is given by dy/dx = (2x₁) / (-2y₁).

The slope of the normal chord can be determined by taking the negative reciprocal of dy/dx:

m₁ = -1 / [dy/dx] = -1 / [(2x₁) / (-2y₁)] = y₁ / x₁

Since the normal chord passes through the focus, its equation can be written as:

y - y₁ = (y₁ / x₁) * (x - x₁) --(3)

Finding the Intersection Point:
To determine the point of intersection of the tangents at the ends of the normal chord, we