Locus of Midpoint of Chord of Contact to Circle

To find the locus of the mid-point of the chord of contact, we proceed as follows:

Let $P(x_1, y_1)$ be any point lying on the given straight line: $4x_1 - 5y_1 = 20 \quad \text{--- (1)}$

The equation of the chord of contact of tangents drawn from $P(x_1, y_1)$ to the circle $x^2 + y^2 = 9$ is given by $T = 0$: $x x_1 + y y_1 = 9 \quad \text{--- (2)}$

Let $M(h, k)$ be the mid-point of this chord of contact. The equation of a chord of the circle with given mid-point $(h, k)$ is represented by $T = S_1$: $x h + y k - 9 = h^2 + k^2 - 9$ $x h + y k = h^2 + k^2 \quad \text{--- (3)}$

Since equations (2) and (3) represent the exact same straight line, their corresponding coefficients must be proportional: $\frac{x_1}{h} = \frac{y_1}{k} = \frac{9}{h^2 + k^2}$

Solving for $x_1$ and $y_1$ in terms of $h$ and $k$: $x_1 = \frac{9h}{h^2 + k^2}$ $y_1 = \frac{9k}{h^2 + k^2}$

Since $(x_1, y_1)$ lies on the line $4x - 5y = 20$ (from equation (1)), we substitute these values into the line's equation: $4\left(\frac{9h}{h^2 + k^2}\right) - 5\left(\frac{9k}{h^2 + k^2}\right) = 20$

Multiplying both sides by $(h^2 + k^2)$: $36h - 45k = 20(h^2 + k^2)$ $20(h^2 + k^2) - 36h + 45k = 0$

Replacing $(h, k)$ with $(x, y)$, the required locus of the mid-point is: $20(x^2 + y^2) - 36x + 45y = 0$

Therefore, the correct option is (A).