Question 826622
Use the point and see what happens:


{{{(4)^2+(2)^2=5*4}}}
{{{16+4=20}}}
{{{20=20}}}
Good so far.


We MUST have radius*radius=20.
The circle is centered at the origin.  The form of the equations tells that.
Radius is {{{2*sqrt(5)}}}.
The equation for the circle is {{{x^2+y^2=20}}}


What have we now?
The center of the circle is (0,0), based on the equation for it; and the radius is {{{sqrt(20)}}}.  Point P(4,2) is a point on the circle and we WANT the endpoint of the diameter other than P.  


We also expect the diameter to contain both P and the origin (0,0).  What is the line for these two points?  The both endpoints of the diameter must be on this line!


Line defined by (0,0) and (4,2):
slope is (1/2).  Obviously runs through origin.
Equation for line is {{{y=(1/2)x}}}.


What points are on this line AND are sqrt(20) distance from the origin (0,0)?  We look for points (x,y), as (x,(1/2)x).  
USE DISTANCE FORMULA.
'
{{{sqrt(20)=sqrt((x-0)^2+((1/2)x-0)^2)}}}
{{{20=x^2+(1/4)x^2}}}
{{{(x^2)(1+1/4)=20}}}
{{{(5/4)x^2=20}}}
{{{x^2=(4/5)20=16}}}
{{{x=0+- 4}}}


We already know about x=4 based on our given point on the circle.  We use the equation for the line {{{y=(1/2)x}}} to find the OTHER endpoint for the diameter.  Use {{{x=-4}}} to find the y value for this endpoint:
{{{y=(1/2)(-4)}}}
{{{y=-2}}}
FINISHED ANSWER: The other endpoint for the diameter at opposite end from (4,2) is (-4,-2).
DONE!


-Most of solution above was modified and corrected-