First case: The Colpitts oscillator is sort of a reverse on the Hartley oscillator. Instead of a tapped coil, it uses two capacitors in series to provide the feedback point.
Because feedback is through the capacitive leg of the LC tank circuit, the total capacitance of this leg is the series combination of C1 and C2, so that
C = (C1*C2)/(C1+C2). The operating frequency is controlled by the tank circuit: ω = 2πf = 1/√(LC)
Because the transistor cannot be biased through the capacitors, we need a separate DC biasing circuit for the transistor itself. That is the purpose of the source resistor in this circuit. Of course, parasitic capacitance in the resistor and the transistor will have a small effect on operating frequency. However, this can be balanced out by careful adjustment of C in the tank circuit.
Second case:(Source is here)
Clapp oscillator is just a modification of Colpitts oscillator. The
only difference is that there is one additional capacitor connected in
series to the inductor in the tank circuit. The main purpose of adding
this additional capacitor C3 is to improve the frequency stability.
The insertion of this additional capacitor C3 prevents the stray
capacitances and other parameters of the transistor from affecting C1
and C2. In variable frequency applications using Clapp oscillator,
the common practice is to make the C1, C2 fixed and C3 is made
variable. While deriving the frequency equation the additional
capacitor must be also taken into consideration and the equation is:
The value of C3 is usually selected much small and so the value of C1
and C2 has less effect on the net effective capacitance. As a result
the equation for frequency can be simplified as