Which of the given sets are connected

Which of the given sets are connected

Which of the following sets are connected?
a. The set $\{f(x, y)\in R:xy = 1\}$
b. The set of all symmetric matrices in $M_n(R).$
c. The set of all orthogonal matrices in $M_n(R).$



My attempt:
a) False ($(x,y)¡æx$ is continuous under which the image is
$R-\{0\}$,disconnected, but connectedness is invariant under continuous
map),
b) True (For $A,B$ symmetric $(1-t)A+tB~¢£~0¡Ât¡Â1$ is symmetric whence
the set is path connected),
c) False ($\det$ is continuous under which the image is $\{-1, 1\}$
disconnected, but connectedness is invariant under continuous map)
Am I correct?