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Step-by-Step Solution
Step 1: Express the given quantities
Let the complex number be
$z = x + iy$.
We are given
$u = \frac{2z + i}{z - ki}$
where
$k > 0$.
Step 2: Simplify the expression for u
Substitute
$z = x + iy$
into
$u$:
$u
= \frac{2(x + iy) + i}{(x + iy) - ki}
= \frac{2x + i(2y + 1)}{x + i(y - k)}.$
To simplify, multiply numerator and denominator by the complex conjugate of the denominator
$x - i(y - k)$:
$u
= \frac{\bigl[2x + i(2y + 1)\bigr]\bigl[x - i(y - k)\bigr]}
{\bigl[x + i(y - k)\bigr]\bigl[x - i(y - k)\bigr]}
= \frac{\text{(Real part)} + i\,\text{(Imag. part)}}{x^2 + (y-k)^2}.$
Though the detailed expansion is given, the key point is that
$u$
can be written in a simplified form where both real and imaginary parts are over the common denominator
$x^2 + (y - k)^2$.
Step 3: Apply Re(u) + Im(u) = 1
We are given:
$\mathrm{Re}(u) + \mathrm{Im}(u) = 1.$
After simplification (using the expanded form of the numerator), this becomes:
$2x^2 + (2y + 1)(y - k) + x + 2kx
= x^2 + (y - k)^2.$
Step 4: Find intersections with the y-axis
On the y-axis,
$x=0$.
So we substitute
$x=0$
into the equation to find the possible y-values of intersection points
$P$
and
$Q$.
With
$x=0$,
the equation simplifies to:
$(2y + 1)\,(y - k) = (y - k)^2.$
Rearrange and simplify:
$2y^2 + y - 2yk - k = y^2 + k^2 - 2ky
\quad\Longrightarrow\quad
y^2 + y - (k + k^2) = 0.
Thus, the y-coordinates
$y_1$
and
$y_2$
of points
$P$
and
$Q$
are the roots of
$y^2 + y - (k + k^2) = 0.$
Step 5: Use properties of quadratic roots
For the equation
$y^2 + y - (k + k^2)=0$,
let its roots be
$y_1$
and
$y_2$.
Then:
$y_1 + y_2 = -\frac{\text{coefficient of }y}{\text{coefficient of }y^2} = -1 \text{ (but note: check the sign carefully)}$
Here, the equation is
$y^2 + y - (k + k^2)=0.$
Sum of roots:
$y_1 + y_2 = -\frac{1}{1} = -1.$
Product of roots:
$y_1\,y_2 = - (k + k^2).$
(Note: The provided solution stated
$y_1 + y_2 = 1,$
but from standard quadratic formula
$y_1 + y_2 = -1.$
We can proceed carefully with the final distance calculation.)
Step 6: Distance PQ on the y-axis
Points
$P$
and
$Q$
are
$(0,\,y_1)$
and
$(0,\,y_2)$.
Hence the distance between them is
$|y_1 - y_2|.$
Using the identity:
$(y_1 - y_2)^2 = (y_1 + y_2)^2 - 4\,y_1\,y_2.$
So,
$|y_1 - y_2|
= \sqrt{\,(y_1 + y_2)^2 - 4\,y_1\,y_2}
= \sqrt{(-1)^2 - 4\,[-(k + k^2)]}
= \sqrt{1 + 4(k + k^2)}
= \sqrt{4k^2 + 4k + 1}
= \sqrt{(2k + 1)^2}
= |2k + 1|.
We are given
$PQ = 5,$
so
$|2k + 1| = 5.$
Step 7: Solve for k
The equation
$|2k + 1| = 5$
gives two possibilities:
$2k + 1 = 5 \quad\Longrightarrow\quad 2k = 4 \quad\Longrightarrow\quad k=2,$
$2k + 1 = -5 \quad\Longrightarrow\quad 2k = -6 \quad\Longrightarrow\quad k=-3.$
But
$k>0$
(by the problem statement). Thus, the only acceptable value is
$k=2$.
Answer: 2
