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.

Question

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. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/bf372ae17f0ca9e242d83b9c4a19740a-.png" /> Given, operatornameRe(z)-2=|z-7+2 i| ⇒ ( x -2)2=( x -7)2+( y +2)2 ⇒ ( y +2)2=10( x -(9/2)) So, the given locus is that of a parabola with directrix x=2 and focus (7,-2). Clearly, minimum PQ =l( L . R )=.10

Given, Re(z)−2=∣z−7+2i∣ ⇒(x−2)2=(x−7)2+(y+2)2 ⇒(y+2)2=10(x−29​) So, the given locus is that of a parabola with directrix x=2 and focus (7,−2). Clearly, minimum PQ=l(L.R)=.10

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