Question

the graph shows the quotient of 12 + 8i and a number. what is the number that 12 + 8i is divided by to yield the number represented by point a on the complex plane? the answer: imaginary 16 12 12 + 8i 8 4 real 4 8 12 16

Explanation:

Step1: Identify point A on complex plane

Point A has real part 2, imaginary part 2, so it is $2 + 2i$.

Step2: Let divisor be $x+yi$, set up equation

Let $\frac{12 + 8i}{x + yi} = 2 + 2i$. Rearrange to solve for $x+yi$:
$x + yi = \frac{12 + 8i}{2 + 2i}$

Step3: Rationalize the denominator

Multiply numerator and denominator by conjugate $2-2i$:
$x + yi = \frac{(12 + 8i)(2 - 2i)}{(2 + 2i)(2 - 2i)}$
Calculate denominator: $(2)^2 - (2i)^2 = 4 - (-4) = 8$
Calculate numerator: $12*2 + 12*(-2i) + 8i*2 + 8i*(-2i) = 24 -24i +16i -16i^2 = 24 -8i +16 = 40 -8i$

Step4: Simplify the fraction

$x + yi = \frac{40 - 8i}{8} = 5 - i$

Answer:

$5 - i$