1. Parabola and its focus

Given parabola: $y^2=20x$

Compare with the standard form $y^2=4ax$, so: $4a=20 \implies a=5$ Hence the focus is: $R=(5,0)$

2. Let the intersection points be $P$ and $Q$

The line is: $y=mx+c$

It intersects the parabola at two points $P,Q$.

Let the $y$-coordinates of $P,Q$ be $y_1,y_2$. Since on the parabola $x=\dfrac{y^2}{20}$, the points are: $P\left(\frac{y_1^2}{20},y_1\right), \quad Q\left(\frac{y_2^2}{20},y_2\right)$

Because both lie on the line $y=mx+c$, $y = m\left(\frac{y^2}{20}\right)+c$ So $y$ satisfies: $my^2-20y+20c=0$

Thus, for roots $y_1,y_2$: $y_1+y_2=\frac{20}{m}, \qquad y_1y_2=\frac{20c}{m}$

3. Use centroid condition

Centroid of triangle $PQR$ is $G(10,10)$.

So: $\left(\frac{x_P+x_Q+x_R}{3},\frac{y_P+y_Q+y_R}{3}\right)=(10,10)$

Since $R=(5,0)$,

From the $y$-coordinate:

$\frac{y_1+y_2+0}{3}=10$ $y_1+y_2=30$

Using $y_1+y_2=\dfrac{20}{m}$, $\frac{20}{m}=30 \implies m=\frac{2}{3}$

Given: $c-m=6$ So: $c=6+\frac{2}{3}=\frac{20}{3}$

4. Check with the $x$-coordinate of centroid

We need: $\frac{x_P+x_Q+5}{3}=10$ $x_P+x_Q=25$

But: $x_P+x_Q=\frac{y_1^2+y_2^2}{20}$ So: $\frac{y_1^2+y_2^2}{20}=25 \implies y_1^2+y_2^2=500$

Now, $(y_1+y_2)^2=y_1^2+y_2^2+2y_1y_2$ $30^2=500+2y_1y_2$ $900=500+2y_1y_2$ $y_1y_2=200$

Also from the quadratic relation: $y_1y_2=\frac{20c}{m}=\frac{20\cdot (20/3)}{2/3}=200$ which is consistent.

5. Find $(PQ)^2$

Coordinates: $P\left(\frac{y_1^2}{20},y_1\right),\quad Q\left(\frac{y_2^2}{20},y_2\right)$

Hence, $PQ^2=\left(\frac{y_1^2-y_2^2}{20}\right)^2+(y_1-y_2)^2$

Factor: $y_1^2-y_2^2=(y_1-y_2)(y_1+y_2)$ So: $PQ^2=(y_1-y_2)^2\left(\frac{(y_1+y_2)^2}{400}+1\right)$

Now, $(y_1-y_2)^2=(y_1+y_2)^2-4y_1y_2=30^2-4\cdot 200=900-800=100$

Therefore, $PQ^2=100\left(\frac{30^2}{400}+1\right)=100\left(\frac{900}{400}+1\right)$ $=100\left(\frac{9}{4}+1\right)=100\cdot \frac{13}{4}=325$

6. Compare with options

$PQ^2=325$ So the correct option is: B

7. Comparison with stored answer

Stored correct answer: B

Our derived answer matches the stored answer.