(c)Given numbers are
${z_1} = 10 + 6i,{z_2} = 4 + 6i$and $z = x + iy$
$amp\left( {\frac{{z – {z_1}}}{{z – {z_2}}}} \right) = \frac{\pi }{4}$==> $amp\left[ {\frac{{(x – 10) + i\,(y – 6)}}{{(x – 4) + i\,(y – 6)}}} \right] = \frac{\pi }{4}$
==> $\frac{{(x – 4)(y – 6) – (y – 6)(x – 10)}}{{(x – 4)(x – 10) + {{(y – 6)}^2}}} = 1$
==> $12y – {y^2} – 72 + 6y = {x^2} – 14x + 40$ …..$(i)$
Now $|z – 7 – 9i|\, = |\,(x – 7) + i(y – 9)|$
==> $\sqrt {{{(x – 7)}^2} + {{(y – 9)}^2}} $ ….$(ii)$
From $(i),$  $({x^2} – 14x + 49) + ({y^2} – 18y + 81) = 18$
==> ${(x – 7)^2} + {(y – 9)^2} = 18$
or ${[{(x – 7)^2} + {(y – 9)^2}]^{1/2}} = {[18]^{1/2}} = 3\sqrt 2 $
$|(x – 7) + i(y – 9)| = 3\sqrt 2 $or $|z – 7 – 9i| = 3\sqrt 2 $.