1. Tardigrade
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  4. Consider the locus of the complex number z in the argand plane given by operatornameRe(z)-2= mid z-7+ 2 i mid. Let P ( z 1) and Q ( z 2) be two complex numbers satisfying the given locus and also satisfying arg ((z1-(2+α i)/z2-(2+α i)))=(π/2)(α ∈ R). Find the minimum value of PQ. [Note: operatornameRe(z) denotes real part of complex number z and i2=-1.

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Q. Consider the locus of the complex number $z$ in the argand plane given by $\operatorname{Re}(z)-2=\mid z-7+$ $2 i \mid$.
Let $P \left( z _1\right)$ and $Q \left( z _2\right)$ be two complex numbers satisfying the given locus and also satisfying $\arg \left(\frac{z_1-(2+\alpha i)}{z_2-(2+\alpha i)}\right)=\frac{\pi}{2}(\alpha \in R)$. Find the minimum value of PQ.
[Note: $\operatorname{Re}(z)$ denotes real part of complex number $z$ and $i^2=-1.$

Complex Numbers and Quadratic Equations

Solution:

image
Given, $\operatorname{Re}(z)-2=|z-7+2 i|$
$\Rightarrow ( x -2)^2=( x -7)^2+( y +2)^2$
$\Rightarrow ( y +2)^2=10\left( x -\frac{9}{2}\right)$
So, the given locus is that of a parabola with directrix $x=2$ and focus $(7,-2)$.
Clearly, minimum $PQ =l( L . R )=$.10