A function f is defined on the complex number by f(z) = (a + ib)z… - Vidyadip

MCQ

A function $f$ is defined on the complex number by $f(z) = (a + ib)z$ , where $a,b \in {\mathbb{R}^ + }$ .This function has the property that the $f-$ image of any point in the complex plane is equidistant from that point and origin. If $|a + bi|= 10$ and ${b^2} = \frac{p}{q}\,;\,p,q \in \mathbb{Z}$ , $gcd(p, q) = 1$ , then $p + q$ is

A
$503$
✓
$403$
C
$405$
D
none of these

✓

Answer

Correct option: B.

$403$

b
$|(a+i b) z-z|=|(a+i b) z|$

$|z||a-1+i b|=|z||a+i b|$

$\Rightarrow \quad a=\frac{1}{2}$

Now $a^{2}+b^{2}=100$

$\Rightarrow \quad b^{2}=\frac{399}{4}$

$\Rightarrow p+q=399+4=403$

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