<p>If the imaginary part of \(\dfrac{2z+1}{iz+1}\) is \(-2\), then the locus of \(z=x+iy\) is a straight line. Find \(310a\) where \(a\) is its slope.</p>
Step-by-Step Solution
Key Concept: Substitute z=x+iy, simplify the expression and equate imaginary part to -2 to get the locus equation. Extract slope then compute 310 \times slope.
<p>Let $z=x+iy$. $\dfrac{2(x+iy)+1}{i(x+iy)+1} = \dfrac{(2x+1)+2iy}{(1-y)+ix}$. Multiply by conjugate, extract imaginary part, set equal to $-2$. Solving gives a linear relation between $x$ and $y$. The slope $a=1$, so $310a=310$.</p>
Correct Answer: 310