If {z_1} = 10 + 6i,{z_2} = 4 + 6i and z is a | vedclass

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Question

(c)Given numbers are
${z_1} = 10 + 6i,{z_2} = 4 + 6i$and $z = x + iy$
$amp\left( {\frac{{z - {z_1}}}{{z - {z_2}}}} \right) = \frac{\pi }{4}$==> $

Vedclass · Accepted Answer

$3\sqrt 2 $

Vedclass · Answer

(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 $.

If ${z_1} = 10 + 6i,{z_2} = 4 + 6i$ and $z$ is a complex number such that $amp\left( {\frac{{z - {z_1}}}{{z - {z_2}}}} \right) = \frac{\pi }{4},$ then the value of $|z - 7 - 9i|$ is equal to

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