1. Fundamental Concept: The Chord of Contact

This problem is a classic application of a powerful concept in coordinate geometry: the chord of contact for a conic section. Understanding this concept is key to solving the problem efficiently and elegantly.

What is a Chord of Contact? Imagine an external point $P(x_1, y_1)$ located outside a conic curve (such as a parabola, circle, ellipse, or hyperbola). If we draw two tangents from this external point $P$ to the conic, these tangents will touch the curve at two distinct points. Let's call these points of tangency $A$ and $B$. The line segment connecting these two points of tangency, $A$ and $B$, is called the chord of contact from the external point $P$.

Equation of the Chord of Contact ($T=0$): A remarkable aspect of conic sections is that the equation of this chord of contact can be found directly without first determining the specific points of tangency or the equations of the individual tangents. If the general equation of a conic is given by $S \equiv Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, and the external point from which the tangents are drawn is $P(x_1, y_1)$, then the equation of the chord of contact is given by $T=0$. The expression $T$ is obtained by making the following specific substitutions into the equation of the conic $S$:

• Replace $x^2$ with $x x_1$
• Replace $y^2$ with $y y_1$
• Replace $xy$ with $\frac{xy_1 + yx_1}{2}$
• Replace $x$ with $\frac{x+x_1}{2}$
• Replace $y$ with $\frac{y+y_1}{2}$
• Constant terms (like $F$) remain unchanged.

2. Problem Analysis and Strategic Approach

The problem statement asks: "The tangents to the curve $y = (x – 2)^2 – 1$ at its points of intersection with the line $x – y = 3$, intersect at the point :".

Let's break down this information to understand the scenario and formulate our strategy:

1. The Curve: We are given the equation $y