A10 ohm resistor is connected in parallel across series combination of 4 ohms and 6 ohms resistors. calculate the amount of heat produced per second, in 4 ohms resistor if 10 joules heat is produced per second in 10 ohms resistor

OK.  We can do this  !

--  We need to know the current through the 4-ohm resistor.

-- But the only thing we know is the power dissipated by the 10-ohm
resistor.  Can we use that to find the current through the 4-ohm resistor ?

You bet we can !

-- The effective resistance of the series combination  is (4+6) = 10 ohms.

-- Both parallel branches have the same resistance, so the current
is the same in both branches.

-- The power dissipated by a resistor is          P = I² · R
For the 10-ohm resistor,

             (10 joules/sec) = (I²) · 10-ohms

                   I = √(10/10)  =  1 ampere

-- The current in the series combination is the same as the current
in the 10-ohm resistor . . . 1 ampere.

-- So the power dissipated by the 4-ohm resistor in the series branch is

                 P = I² · R  =  (1 amp)² · (4 ohms)  =  4 watts.