Understanding Gravitational Force
When two masses exert a gravitational force on a third mass, there exists a point where the net gravitational force experienced by the third mass becomes zero. In this case, we have two spherical balls with masses of 1 kg and 4 kg, separated by a distance of 12 cm.

Configuration of the Problem
- Mass of ball A (1 kg) = 1 kg
- Mass of ball B (4 kg) = 4 kg
- Distance between ball A and ball B = 12 cm
Let the distance from the 1 kg mass (ball A) to the point where the gravitational force becomes zero be \( x \). Thus, the distance from the 4 kg mass (ball B) to this point will be \( (12 - x) \) cm.

Setting Up the Equation
The gravitational force exerted by mass A on the point is given by:
\[
F_A = \frac{G \cdot 1 \cdot m}{x^2}
\]
The gravitational force exerted by mass B on the point is:
\[
F_B = \frac{G \cdot 4 \cdot m}{(12 - x)^2}
\]
Setting these two forces equal to each other for equilibrium:
\[
\frac{1}{x^2} = \frac{4}{(12 - x)^2}
\]

Cross Multiplying and Solving
Cross-multiplying gives:
\[
(12 - x)^2 = 4x^2
\]
Expanding both sides:
\[
144 - 24x + x^2 = 4x^2
\]
Rearranging terms:
\[
3x^2 - 24x + 144 = 0
\]
Dividing by 3:
\[
x^2 - 8x + 48 = 0
\]

Finding the Roots
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1, b = -8, c = 48 \).
- The discriminant is \( (-8)^2 - 4 \cdot 1 \cdot 48 = 64 - 192 = -128 \).
Since the discriminant is negative, this means there is no real solution for \( x \) within the range of the two masses; thus, the gravitational force cannot become zero at any point between or outside the two balls.