To solve for $p$, we are given that the line $3x + 4y - p = 0$ is a tangent to the circle $x^2 + y^2 = 16$.

Step 1: Equation of the circle

The equation of the circle is: $x^2 + y^2 = 16$ This is a circle with center at $(0, 0)$ and radius $r = \sqrt{16} = 4$.

Step 2: Equation of the line

The equation of the line is: $3x + 4y - p = 0$ We can rewrite this as: $3x + 4y = p$ This line is in the form $Ax + By + C = 0$, where $A = 3$, $B = 4$, and $C = -p$.

Step 3: Condition for tangency

For a line to be tangent to a circle, the perpendicular distance from the center of the circle to the line must be equal to the radius of the circle.

The formula for the perpendicular distance $d$ from a point $(x_1, y_1)$ to the line $Ax + By + C = 0$ is given by: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ In this case, the center of the circle is $(0, 0)$, so the distance from the center to the line is: $d = \frac{|3(0) + 4(0) - p|}{\sqrt{3^2 + 4^2}} = \frac{| - p |}{\sqrt{9 + 16}} = \frac{|p|}{5}$ Since the line is a tangent, this distance must be equal to the radius of the circle, which is 4. Thus: $\frac{|p|}{5} = 4$ Solving for $p$: $|p| = 20$

Step 4: Final answer

Since $|p| = 20$, $p$ can be either $20$ or $-20$.

Thus, the value of $p$ is $\boxed{20}$ or $\boxed{-20}$.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the general formula for the perpendicular distance between a point and a line?
2. How do you find the equation of a tangent line to a circle at a given point?
3. What is the geometric interpretation of a line being tangent to a circle?
4. How do you derive the equation of a circle given its center and radius?
5. What are the conditions for two lines to be perpendicular or parallel?

Tip: When working with tangents, always check that the perpendicular distance from the center of the circle to the line equals the radius.