What An Eclipse!

The problem says that it illuminates 23% of the larger sphere. As the figure suggests, and as explained that the light rays seem to emanate from the center of the light source, some of these rays will be tangential to the larger sphere, such the area illuminated resembles a spherical cap . Using its formula for its area, we can determine the height of this cap.

$2\pi rh = \frac{23}{100} 4 \pi r^2$

$h = \frac{23}{50}r$

Here, $r = 6000$ .

Now, if we view slice these figures vertically oriented with the sliced half facing us, we will see this figure.

We know that tangent lines perpendicularly meet circles at their radii. If we also connect two tangent lines to the circle from the same point (which in this case is the center of the light source), we can now draw relationships via similar triangles.

Let $D$ be the point of intersection of the segment connecting the points of tangency of $B$ at circle $A$ and segment $AB$ .

We can see from the figure that segment $AB = r + x + 800$ where $x$ is the nearest distance between the two spheres.

We know that $\Delta ACB$ is a right triangle and so is $\Delta ADC$ . Here we can deduce that

$\frac {AC}{AB} = \frac {AD}{AC}$

$\frac {r}{r+x+800} = \frac{r-h}{r}$

So,

$d = \frac {r^2}{r-h} - r - 800$

Using the fact that $h = \frac{23}{50}r$ ,

$d = \frac{23}{27}r - 800$

so $d \approx 4311.111 km$

The curved surface area of a spherical cap can be calculated as

                                         A=2πr^2 (1-cos θ)

Where r is the radius and θ the angle measured at the spears center between the center of the cap area and the circular cap edge on the spears surface.

The total area of a shear is calculated

                                             A=4πr^2

23% of the total area is in the cap

                              (2πr^2 (1-cos θ))/(4πr^2 )= .23

                                   cos θ= .54

                                   cos θ = op /hyp = 6000 /AB

                                    AB =6000/.54 = 11,111.111

Distance between the center of the two spherical bodies is 11,111 km

subtract the distance to surface from center of the bodies

                                11111- 6800 = 4311.111 km