Let there be an 
-axis passing through points 
and 
. Let 
at point and 
at point .
Now consider an arbitrary point 
on the -axis.
The magnitude 
of electric field strength due to the point charge 
at point is
,
whereas the magnitude 
of electric field strength due to the point charge 
at point is
.
The point(s) at which 
is found by solving for 
:
There are two points 
and 
at which 
.
The necessary and sufficient condition for 
is that they are equal in magnitude: 
and opposite in sign: 
,
i.e.,
i. and ii. the directions of 
and 
are opposite to each other.
Suffice it to check whether 
and 
at the above-mentioned two points are in opposite direction.
The first candidate , where 
, is in between point and point . If a positive test charge were placed there, both the electric field strength due to at and due to at would point to the right. That is, this point fails to meet our requirement ii.
The second candidate , where 
, is to the left of point . If a positive test charge were placed at , the electric field strength there due to at would point to the left. Whereas the electric field strength there due to at would point to the right.
We conclude that the resultant electric field strength is zero at only one point, i.e., the point .
And the answer is C.