A light ray emerging from the point source placed at P(1, 3) is reflected at a point Q in the axis of x. If the reflected ray passes through the point R(6, 7), then the abscissa of Q is:

A light ray starts at point P(1, 3) and reflects off the x-axis at point Q, then passes through R(6, 7). Since Q lies on the x-axis, its coordinates are (x, 0), and we need to find the abscissa x.

When a ray reflects off a surface, the angle of incidence equals the angle of reflection. For reflection over the x-axis, the mirror image of point R(6, 7) is R'(6, -7). The path from P to Q to R is equivalent to a straight line from P to R' because reflection preserves the path length and direction in the mirrored space. Therefore, Q is the point where the line segment joining P(1, 3) and R'(6, -7) intersects the x-axis.

To find the equation of the line passing through P(1, 3) and R'(6, -7), first calculate the slope m:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-7 - 3}{6 - 1} = \frac{-10}{5} = -2 $$

Using the point-slope form with point P(1, 3):

$$ y - y_1 = m(x - x_1) $$

$$ y - 3 = -2(x - 1) $$

$$ y - 3 = -2x + 2 $$

$$ y = -2x + 2 + 3 $$

$$ y = -2x + 5 $$

This line intersects the x-axis where y = 0:

$$ 0 = -2x + 5 $$

$$ 2x = 5 $$

$$ x = \frac{5}{2} $$

Thus, the abscissa of Q is $$ \frac{5}{2} $$.

Alternatively, using the reflection property directly, let Q be (x, 0). The incident ray is from P(1, 3) to Q(x, 0), so its direction vector is (x - 1, -3). The reflected ray is from Q(x, 0) to R(6, 7), so its direction vector is (6 - x, 7). For reflection over the x-axis, the x-component of the direction remains the same, and the y-component reverses. Thus, the reflected direction should be proportional to (x - 1, 3) (since the incident y-component is -3, reversing gives +3). Therefore, the vectors (6 - x, 7) and (x - 1, 3) must be parallel, meaning their components are proportional:

$$ \frac{6 - x}{x - 1} = \frac{7}{3} $$

Cross-multiplying:

$$ 3(6 - x) = 7(x - 1) $$

$$ 18 - 3x = 7x - 7 $$

$$ 18 + 7 = 7x + 3x $$

$$ 25 = 10x $$

$$ x = \frac{25}{10} = \frac{5}{2} $$

Both methods confirm the abscissa of Q is $$ \frac{5}{2} $$.

Hence, the correct answer is Option D.