a,
$C\in d: x+y=4\Leftrightarrow y=-x-4$
$\to C(t;-t-4)$
$CA=\sqrt{(t-1)^2+(-t-5)^2}=\sqrt{(t-1)^2+(t+5)^2}=\sqrt{2t^2+8t+26}$
$CB=\sqrt{(t+1)^2+(-t-7)^2}=\sqrt{(t+1)^2+(t+7)^2}=\sqrt{2t^2+16t+50}$
$CA=CB\to CA^2=CB^2$
$\to 2t^2+8t+26=2t^2+16t+50$
$\to t=-3$
$\to C(-3;-1)$
b,
$ABCD$ là HBH
$\to \vec{AB}=\vec{DC}$
$\vec{AB}(-2;2)$
$\to D(-3+2; -1-2)=(-1;-3)$
$\vec{n_{AB}}(2;2)=(1;1)$
$\to AB: x-1+y-1=0\to x+y-2=0$
$\to h=d(C;AB)=\dfrac{|-3-1-2|}{\sqrt{1^2+1^2}}=3\sqrt2$
$AB=\sqrt{2^2+2^2}=2\sqrt2$
$\to S_{ABCD}=2\sqrt2.h=12$