Locus of Tangent Intersections for Chords Through Fixed Point

To determine the curve on which the points $E_3$, $F_3$, and $G_3$ lie, we can use the concept of poles and polars with respect to a circle.

1. General Property

For a circle $S: x^2 + y^2 = r^2$, if a variable chord passes through a fixed point $P_0(x_0, y_0)$, then the point of intersection of the tangents drawn at the ends of the chord is the pole of that chord.

By the principle of polar reciprocity, since every such chord passes through the fixed point $P_0(x_0, y_0)$, the locus of the intersection points of these tangents (the poles) is the polar of the point $P_0(x_0, y_0)$ with respect to the circle $S$.

The equation of the polar of a point $P_0(x_0, y_0)$ with respect to $x^2 + y^2 = r^2$ is given by: $T = 0 \implies x x_0 + y y_0 = r^2$

Given:

• Circle equation $S: x^2 + y^2 = 4$ (so $r^2 = 4$)
• Fixed point $P_0(x_0, y_0) = (1, 1)$

Thus, the locus of the intersection of the tangents for any chord passing through $P_0(1, 1)$ is: $x(1) + y(1) = 4 \implies x + y = 4$

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2. Verification using the Given Points

We can explicitly calculate the coordinates of $E_3$, $F_3$, and $G_3$:

1. Point $E_3$:

   • Chord $E_1E_2$ passes through $P_0(1, 1)$ and is parallel to the $x$-axis, so its equation is $y = 1$.
   • Let $E_3 = (h_1, k_1)$. The chord of contact for $E_3$ with respect to $x^2 + y^2 = 4$ is $h_1 x + k_1 y = 4$.
   • Comparing $h_1 x + k_1 y = 4$ with $y = 1$ (i.e., $0 \cdot x + 4y = 4$), we get: $h_1 = 0, \quad k_1 = 4 \implies E_3 = (0, 4)$

2. Point $F_3$:

   • Chord $F_1F_2$ passes through $P_0(1, 1)$ and is parallel to the $y$-axis, so its equation is $x = 1$.
   • Let $F_3 = (h_2, k_2)$. The chord of contact for $F_3$ is $h_2 x + k_2 y = 4$.
   • Comparing $h_2 x + k_2 y = 4$ with $x = 1$ (i.e., $4x + 0 \cdot y = 4$), we get: $h_2 = 4, \quad k_2 = 0 \implies F_3 = (4, 0)$

3. Point $G_3$:

   • Chord $G_1G_2$ passes through $P_0(1, 1)$ with slope $-1$, so its equation is: $y - 1 = -1(x - 1) \implies x + y = 2$
   • Let $G_3 = (h_3, k_3)$. The chord of contact for $G_3$ is $h_3 x + k_3 y = 4$.
   • Comparing $h_3 x + k_3 y = 4$ with $2x + 2y = 4$, we get: $h_3 = 2, \quad k_3 = 2 \implies G_3 = (2, 2)$

3. Conclusion

Testing the points $E_3(0, 4)$, $F_3(4, 0)$, and $G_3(2, 2)$ in the line equation $x + y = 4$:

• $E_3(0, 4) \implies 0 + 4 = 4$
• $F_3(4, 0) \implies 4 + 0 = 4$
• $G_3(2, 2) \implies 2 + 2 = 4$

All three points lie on the straight line $x + y = 4$.

Correct Option: (A) $x + y = 4$