Length of Chord of Circle Intersected by Line

To find the square of the length of the chord of the circle on the line $x = 1$, we proceed step-by-step:

Step 1: Identify the center and radius of the circle The general equation of the given circle is: $x^2 + y^2 + 2gx + 2fy + 25 = 0$

The center of the circle is $C(-g, -f)$. Let $h = -g$ and $k = -f$, so the center is $C(h, k)$. Since the center lies in the first quadrant, we have $h > 0$ and $k > 0$.

The radius $R$ of the circle is given by: $R = \sqrt{g^2 + f^2 - 25} = \sqrt{h^2 + k^2 - 25}$ $R^2 = h^2 + k^2 - 25 \quad \text{--- (1)}$

Step 2: Use the line equation for the center The center $C(h, k)$ lies on the line $2x - y = 4$: $2h - k = 4 \implies k = 2h - 4 \quad \text{--- (2)}$

Step 3: Determine the radius using the area of the equilateral triangle An equilateral triangle of side length $s$ inscribed in a circle of radius $R$ satisfies $s = R\sqrt{3}$. The area $A$ of this triangle is: $A = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} (R\sqrt{3})^2 = \frac{3\sqrt{3}}{4} R^2$

Given that $A = 27\sqrt{3}$: $\frac{3\sqrt{3}}{4} R^2 = 27\sqrt{3} \implies R^2 = 36 \implies R = 6$

Step 4: Solve for $h$ and $k$ Substitute $R^2 = 36$ into equation (1): $h^2 + k^2 - 25 = 36 \implies h^2 + k^2 = 61 \quad \text{--- (3)}$

Substitute equation (2) into equation (3): $h^2 + (2h - 4)^2 = 61$ $h^2 + 4h^2 - 16h + 16 = 61$ $5h^2 - 16h - 45 = 0$

Factoring the quadratic equation: $(5h + 9)(h - 5) = 0$

Since $h > 0$, we take $h = 5$. Using equation (2), $k = 2(5) - 4 = 6$. Thus, the center of the circle is $C(5, 6)$ and the radius is $R = 6$.

Step 5: Find the length of the chord on $x = 1$ The equation of the circle with center $C(5, 6)$ and radius $R = 6$ is: $(x - 5)^2 + (y - 6)^2 = 36$

The perpendicular distance $d$ from the center $C(5, 6)$ to the vertical line $x = 1$ is: $d = |5 - 1| = 4$

The length $L$ of the chord intercepted by the circle on the line $x = 1$ is given by: $L = 2\sqrt{R^2 - d^2}$ $L = 2\sqrt{36 - 4^2} = 2\sqrt{36 - 16} = 2\sqrt{20}$

The square of the length of the chord is: $L^2 = (2\sqrt{20})^2 = 4 \times 20 = 80$