$(a)$ Discuss the concept of hybridization. What are its different types in a carbon atom.
$(b)$ What is the type of hybridization of carbon atoms marked with a star in the following structures:
$(a)$ $\overset{*}{C}H_2 = CH - C(=O)OH$
$(b)$ $CH_3 - \overset{*}{C}H_2 - OH$
$(c)$ $CH_3 - CH_2 - \overset{*}{C}H = O$
$(d)$ $\overset{*}{C}H_3 - CH = CH - CH_3$
$(e)$ $CH_3 - \overset{*}{C} \equiv CH$

(A) Hybridization: The process of intermixing of atomic orbitals of slightly different energy to produce an equal number of new hybrid orbitals of equivalent energy and identical shape.
Types of hybridization in carbon:
$1.$ $sp^3$ hybridization: Carbon is bonded to $4$ other atoms by single bonds (tetrahedral geometry).
$2.$ $sp^2$ hybridization: Carbon is bonded to $3$ other atoms,with one double bond (trigonal planar geometry).
$3.$ $sp$ hybridization: Carbon is bonded to $2$ other atoms,with one triple bond or two double bonds (linear geometry).
Identification of hybridization for starred carbons:
$(a)$ $\overset{*}{C}H_2 = CH - C(=O)OH$: The starred carbon is bonded to $2$ hydrogens and $1$ carbon via a double bond. It has $3$ sigma bonds. Hybridization: $sp^2$.
$(b)$ $CH_3 - \overset{*}{C}H_2 - OH$: The starred carbon is bonded to $2$ hydrogens,$1$ carbon,and $1$ oxygen. It has $4$ sigma bonds. Hybridization: $sp^3$.
$(c)$ $CH_3 - CH_2 - \overset{*}{C}H = O$: The starred carbon is bonded to $1$ hydrogen,$1$ carbon,and $1$ oxygen via a double bond. It has $3$ sigma bonds. Hybridization: $sp^2$.
$(d)$ $\overset{*}{C}H_3 - CH = CH - CH_3$: The starred carbon is bonded to $3$ hydrogens and $1$ carbon via single bonds. It has $4$ sigma bonds. Hybridization: $sp^3$.
$(e)$ $CH_3 - \overset{*}{C} \equiv CH$: The starred carbon is bonded to $1$ carbon via a single bond and $1$ carbon via a triple bond. It has $2$ sigma bonds. Hybridization: $sp$.