A blue sphere is fixed at the origin with positive charge q; a red sphere fixed at the point (d1,0) with unknown charge qred; yellow sphere fixed at the point (d2,cos (theta),-d2 sin (theta))with unknown charge qyellow.
The x component of position vector of yellow sphere is positive (d2cos (theta)).
The y component of position vector of yellow sphere is negative ( -d2sin (theta)).
The net electric force on the blue sphere is observed to be F(vector)=(0,-F), where F>0.
As q'blue' and q'red' are on x - axis and the force between them will have no component along y - axis
The net force on charged blue sphere is along negative y - axis and the y component of position vector of yellow sphere is negative ( -d2sin (theta)).
Hence the yellow sphere must be attracting blue sphere,which means,nature of charge on yellow sphere is opposite to nature of charge on blue sphere.
Charge on blue sphere is positive,hence charge on yellow sphere is negative
A) The sign of the charge on the yellow sphere is negative
B) As the component of net force on x - axis is zero,the x component of force on blue due to yellow must be equal and opposite to force on blue due to red
In other words, as yellow attracts blue and red must repel blue.
Hence nature of charge on q'red' is opposite to nature of charge on q'yellow'
Charge on yellow sphere is negative ,hence charge on red is positive
B) The sign of the charge on the red sphere is
As x component of resultant force is zero,
Fx,yellow cos theta = Fx,red
[ k(2q)q/(d2)^2] cos theta =kqred q /(d1)^2
(2q)/(d2)^2] cos theta =qred /(d1)^2
[(2q) (d1)^2 /(d2)^2] cos theta =qred
q red = (2q)* cos theta* [d1 /d2]^2
the charge Qred on the red sphere is (2q)* cos theta* [d1 /d2]^2