For the circle

$$ x^2+y^2+2gx+2fy+25=0, $$

the centre is

$$ (-g,-f) $$

and radius is

$$ r=\sqrt{g^2+f^2-25}. $$

We are given that the centre lies in the first quadrant, so

$$ (-g,-f) $$

is in the first quadrant.

Also, the centre lies on the line

$$ 2x-y=4. $$

So if the centre is $(-g,-f)$, then

$$ 2(-g)-(-f)=4 $$

that is,

$$ -2g+f=4. $$

Now use the information about the equilateral triangle inscribed in the circle.

For an equilateral triangle of side $a$,

$$ \text{Area}=\frac{\sqrt{3}}{4}a^2. $$

Given area is

$$ 27\sqrt{3}. $$

So,

$$ \frac{\sqrt{3}}{4}a^2=27\sqrt{3} $$

which gives

$$ a^2=108 \quad \Rightarrow \quad a=6\sqrt{3}. $$

For an equilateral triangle inscribed in a circle, the circumradius is

$$ R=\frac{a}{\sqrt{3}}. $$

Hence,

$$ r=\frac{6\sqrt{3}}{\sqrt{3}}=6. $$

Therefore,

$$ r^2=36. $$

So,

$$ g^2+f^2-25=36 $$

which gives

$$ g^2+f^2=61. $$

Now solve the system

$$ -2g+f=4 $$

and

$$ g^2+f^2=61. $$

From the first equation,

$$ f=4+2g. $$

Substitute into the second:

$$ g^2+(4+2g)^2=61 $$

$$ g^2+16+16g+4g^2=61 $$

$$ 5g^2+16g-45=0. $$

Solve:

$$ 5g^2+16g-45=0. $$

Discriminant:

$$ \Delta=16^2-4\cdot 5\cdot(-45)=256+900=1156=34^2. $$

Thus,

$$ g=\frac{-16\pm 34}{10}. $$

So,

$$ g=\frac{18}{10}=\frac95 \quad \text{or} \quad g=\frac{-50}{10}=-5. $$

Now centre is $(-g,-f)$ and must be in the first quadrant.

If $g=\frac95$, then

$$ f=4+2\cdot \frac95=\frac{38}{5}, $$

so centre is

$$ \left(-\frac95,-\frac{38}{5}\right), $$

not in first quadrant.

Hence reject.

If $g=-5$, then

$$ f=4+2(-5)=-6. $$

So centre is

$$ (5,6), $$

which is in the first quadrant.

Thus the circle is

$$ x^2+y^2-10x-12y+25=0, $$

with centre $(5,6)$ and radius $6$.

Now we need the chord cut by the line

$$ x=1. $$

Distance of the line $x=1$ from the centre $(5,6)$ is

$$ |5-1|=4. $$

If a chord is at distance $d$ from the centre in a circle of radius $r$, then chord length is

$$ 2\sqrt{r^2-d^2}. $$

So here chord length is

$$ 2\sqrt{6^2-4^2}=2\sqrt{36-16}=2\sqrt{20}=4\sqrt{5}. $$

Therefore, the square of the chord length is

$$ (4\sqrt{5})^2=80. $$

Hence, the required answer is

$$ \boxed{80}. $$

Circle

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