To find the value of |a|, we need to understand the properties of the hyperbola and the concept of tangents.

Properties of the Hyperbola:
The given equation x^2/9 - y^2/16 = 1 represents a hyperbola. The standard form of a hyperbola is (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h,k) represents the center of the hyperbola, and a and b are the lengths of the transverse and conjugate axes, respectively.

In this case, we can rewrite the equation as (x-0)^2/9 - (y-0)^2/16 = 1, which corresponds to a hyperbola centered at the origin (0,0) with a transverse axis of length 2a = 6 and a conjugate axis of length 2b = 8. Therefore, a = 3 and b = 4.

Concept of Tangents:
A tangent to a hyperbola is a straight line that intersects the hyperbola at exactly one point. Tangents can be drawn to a hyperbola from an external point, and there can be two distinct tangents from that point.

Finding the Tangents:
Let's consider the point (a, 2) on the hyperbola. We need to find the two distinct tangents that can be drawn from this point.

Step 1: Substitute the point coordinates into the equation of the hyperbola.
(a^2)/9 - (2^2)/16 = 1

Simplifying this equation, we get:
(a^2)/9 - 4/16 = 1
(a^2)/9 - 1/4 = 1

Step 2: Multiply throughout by 36 to eliminate the denominators.
4(a^2) - 9 = 36

Step 3: Rearrange the equation to standard form.
4(a^2) = 36 + 9
4(a^2) = 45

Step 4: Divide both sides by 4 to isolate a^2.
a^2 = 45/4

Step 5: Take the square root of both sides.
|a| = √(45/4)
|a| = √(9*5/4)
|a| = (3/2)√5

Therefore, the modulus of a is (3/2)√5.

In conclusion, the modulus of a, |a|, is equal to (3/2)√5.