If t and c are two complex numbers such that -t- - -c- -t- - 1 and z - at-bt-c- z-x - iy- then locus of z is a-an where a- b are complex numbers

Z = at+bt c⇒ t = b + czz a Now, |t| = 1 ⇒ | b+czz a |=1 ⇒ |z + bc | |z a|= 1|c| (≠ 1 as |c| ≠ |t|) ⇒ Locus of z is a circle, since we know that nbsp;| z z ...

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Byju's Answer

Standard XI

Mathematics

Equation of Circle in Complex Form

If t and c ar...

Question

If $t$ and $c$ are two complex numbers such that $| t | \neq | c |, | t | = 1$ and $z = \frac{a t + b}{t - c}, z = x + i y$ , then locus of $z$ is a/an (where $a, b$ are complex numbers)

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Solution

The correct option is C circle
$z = \frac{a t + b}{t - c} \Rightarrow t = \frac{b + c z}{z - a}$
Now,
$| t | = 1$
$\Rightarrow ∣ ∣ ∣ \frac{b + c z}{z - a} ∣ ∣ ∣ = 1$
$\Rightarrow \frac{∣ ∣ ∣ z + \frac{b}{c} ∣ ∣ ∣}{| z - a |} = \frac{1}{| c |} (\neq 1 as | c | \neq | t |)$
$\Rightarrow$ Locus of $z$ is a circle, since we know that $∣ ∣ ∣ \frac{z - z_{1}}{z - z_{2}} ∣ ∣ ∣ = k$ denotes a circle, if $k \neq 1 .$

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If t and c are two complex numbers such that |t|≠|c|,|t|=1 and z=at+bt−c,z=x+iy, then locus of z is a/an (where a,b are complex numbers)

The correct option is C circle z=at+bt−c⇒t=b+czz−a Now, |t|=1 ⇒∣∣∣b+czz−a∣∣∣=1 ⇒∣∣∣z+bc∣∣∣|z−a|=1|c| (≠1 as |c|≠|t|) ⇒ Locus of z is a circle, since we know that ∣∣∣z−z1z−z2∣∣∣=k denotes a circle, if k≠1.

If $t$ and $c$ are two complex numbers such that $| t | \neq | c |, | t | = 1$ and $z = \frac{a t + b}{t - c}, z = x + i y$ , then locus of $z$ is a/an (where $a, b$ are complex numbers)