To determine the value of \( r \) for which the equation \( |x^2 - 4x - 13| = r \) has exactly three distinct real roots, we analyze two cases.
This equation can be rewritten as \( x^2 - 4x - (13 + r) = 0 \). For this quadratic equation to have real roots, its discriminant must be non-negative: \( \Delta = (-4)^2 - 4(1)(-(13 + r)) \geq 0 \). This simplifies to \( 16 + 4(13 + r) \geq 0 \), which further simplifies to \( 16 + 52 + 4r \geq 0 \), or \( 68 + 4r \geq 0 \). Thus, \( 4r \geq -68 \), leading to \( r \geq -17 \).
This equation can be rewritten as \( x^2 - 4x - (13 - r) = 0 \). For this quadratic equation to have real roots, its discriminant must be non-negative: \( \Delta = (-4)^2 - 4(1)(-(13 - r)) \geq 0 \). This simplifies to \( 16 + 4(13 - r) \geq 0 \), which further simplifies to \( 16 + 52 - 4r \geq 0 \), or \( 68 - 4r \geq 0 \). Thus, \( 68 \geq 4r \), leading to \( r \leq 17 \).
For the original equation \( |x^2 - 4x - 13| = r \) to have exactly three distinct real roots, one of the quadratic equations must have exactly one real root (a double root), and the other must have two distinct real roots. This occurs when the vertex of the parabola \( y = x^2 - 4x - 13 \) lies on the line \( y = r \) or \( y = -r \). The x-coordinate of the vertex is \( x = -(-4)/(2 \cdot 1) = 2 \). The y-coordinate of the vertex is \( y = (2)^2 - 4(2) - 13 = 4 - 8 - 13 = -17 \). Therefore, the vertex is at \( (2, -17) \). For exactly three distinct real roots, \( r \) must be the absolute value of the y-coordinate of the vertex, which is \( r = |-17| = 17 \). In this scenario, \( x^2 - 4x - 13 = -17 \) yields \( x^2 - 4x + 4 = 0 \), or \( (x-2)^2 = 0 \), giving a double root at \( x=2 \). Simultaneously, \( x^2 - 4x - 13 = 17 \) yields \( x^2 - 4x - 30 = 0 \), which has two distinct real roots since its discriminant \( 16 - 4(1)(-30) = 16 + 120 = 136>0 \).
The value of \( r \) for which the equation has exactly three distinct real roots is \( \boxed{17} \).