Key Concepts and Formulas

• Complex Number Representation: A complex number $z$ can be represented as $z = x + iy$, where $x$ is the real part (Re(z)) and $y$ is the imaginary part (Im(z)), and $i = \sqrt{-1}$.
• Complex Conjugate: The complex conjugate of $z = x + iy$ is $\bar{z} = x - iy$. The product of a complex number and its conjugate is a real number: $z\bar{z} = x^2 + y^2$.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the Cartesian plane is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. The distance between two points $(0, y_1)$ and $(0, y_2)$ on the y-axis is $|y_2 - y_1|$.

Step-by-Step Solution

Step 1: Express $u$ in terms of $x$ and $y$

We are given $u = \frac{2z + i}{z - ki}$ and $z = x + iy$. We need to substitute the expression for $z$ into $u$ and then rationalize the denominator to separate the real and imaginary parts.

$u = \frac{2(x + iy) + i}{(x + iy) - ki} = \frac{2x + 2iy + i}{x + iy - ki} = \frac{2x + i(2y + 1)}{x + i(y - k)}$

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, which is $x - i(y - k)$:

$u = \frac{2x + i(2y + 1)}{x + i(y - k)} \cdot \frac{x - i(y - k)}{x - i(y - k)}$

Step 2: Simplify the expression for $u$

Now, expand the numerator and the denominator:

Numerator: $(2x + i(2y + 1))(x - i(y - k)) = 2x^2 - 2xi(y - k) + xi(2y + 1) - i^2(2y + 1)(y - k)$ Since $i^2 = -1$, $= 2x^2 - 2ixy + 2ixk + 2ixy + ix + (2y + 1)(y - k)$ $= 2x^2 + (2y^2 - 2ky + y - k) + i(x + 2kx)$ $= (2x^2 + 2y^2 - 2ky + y - k) + i(x + 2kx)$

Denominator: $(x + i(y - k))(x - i(y - k)) = x^2 - i^2(y - k)^2 = x^2 + (y - k)^2$

So, $u = \frac{2x^2 + 2y^2 - 2ky + y - k}{x^2 + (y - k)^2} + i\frac{x + 2kx}{x^2 + (y - k)^2}$

Step 3: Identify Re(u) and Im(u)

From the expression for $u$, we have: $Re(u) = \frac{2x^2 + 2y^2 - 2ky + y - k}{x^2 + (y - k)^2}$ $Im(u) = \frac{x + 2kx}{x^2 + (y - k)^2} = \frac{x(1 + 2k)}{x^2 + (y - k)^2}$

Step 4: Apply the condition Re(u) + Im(u) = 1

We are given that $Re(u) + Im(u) = 1$. Substituting the expressions for $Re(u)$ and $Im(u)$, we get:

$\frac{2x^2 + 2y^2 - 2ky + y - k}{x^2 + (y - k)^2} + \frac{x(1 + 2k)}{x^2 + (y - k)^2} = 1$ Combining the fractions: $\frac{2x^2 + 2y^2 - 2ky + y - k + x + 2kx}{x^2 + (y - k)^2} = 1$ Multiplying both sides by the denominator: $2x^2 + 2y^2 - 2ky + y - k + x + 2kx = x^2 + (y - k)^2$ $2x^2 + 2y^2 - 2ky + y - k + x + 2kx = x^2 + y^2 - 2ky + k^2$ $x^2 + y^2 + x + 2kx + y - k - k^2 = 0$

Step 5: Use the intersection with the y-axis (x = 0)

The curve intersects the y-axis, so we set $x = 0$: $(0)^2 + y^2 + (0) + 2k(0) + y - k - k^2 = 0$ $y^2 + y - k - k^2 = 0$

Step 6: Solve for y and find the distance PQ

Let $y_1$ and $y_2$ be the roots of the quadratic equation $y^2 + y - (k^2 + k) = 0$. The distance between the points P and Q is given by $|y_1 - y_2| = 5$.

Using Vieta's formulas, we know that $y_1 + y_2 = -1$ and $y_1 y_2 = -(k^2 + k)$.

We also know that $(y_1 - y_2)^2 = (y_1 + y_2)^2 - 4y_1y_2$. Therefore, $(y_1 - y_2)^2 = (-1)^2 - 4(-(k^2 + k)) = 1 + 4k^2 + 4k$ Since $|y_1 - y_2| = 5$, we have $(y_1 - y_2)^2 = 25$. $1 + 4k^2 + 4k = 25$ $4k^2 + 4k - 24 = 0$ $k^2 + k - 6 = 0$ $(k + 3)(k - 2) = 0$ So, $k = -3$ or $k = 2$.

Step 7: Apply the constraint k > 0

Since $k > 0$, we have $k = 2$.

Common Mistakes & Tips

• Rationalizing the denominator: Make sure to multiply both the numerator and the denominator by the conjugate of the denominator.
• Simplifying the equation: Carefully expand and simplify the expressions to avoid errors.
• Dividing by a factor: Avoid dividing by a factor that could be zero, as you might lose a solution. Instead, factor and set each factor to zero.
• Using Vieta's formulas: For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is $-b/a$ and the product of the roots is $c/a$.
• Checking constraints: Always remember to check if your solution satisfies all the given constraints.

Summary

We started by expressing the complex number $u$ in terms of its real and imaginary parts. Then, we used the given condition $Re(u) + Im(u) = 1$ to obtain an equation. By using the fact that the curve intersects the y-axis, we set $x = 0$ and found a quadratic equation in terms of $y$. We then used Vieta's formulas and the given distance $PQ = 5$ to solve for $k$. Finally, we applied the constraint $k > 0$ to find the correct value of $k$.

The final answer is \boxed{2}, which corresponds to option (A).