Question

a point of the form 25 +bi is 37 units from 13 - 31i. what is the value of b?
○ 4
○ 6.6
○ 36.6
○ 40.6

Explanation:

Step1: Use distance formula for complex numbers

The distance between two complex numbers $z_1 = x_1 + y_1i$ and $z_2 = x_2 + y_2i$ is $d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$. Substitute $z_1=25+bi$, $z_2=13-31i$, $d=37$:
$$37 = \sqrt{(25-13)^2 + (b - (-31))^2}$$

Step2: Simplify the squared terms

Calculate the real part difference and expand the imaginary part:
$$37 = \sqrt{12^2 + (b+31)^2}$$
$$37 = \sqrt{144 + (b+31)^2}$$

Step3: Square both sides to eliminate root

$$37^2 = 144 + (b+31)^2$$
$$1369 = 144 + (b+31)^2$$

Step4: Isolate the squared binomial

Subtract 144 from both sides:
$$(b+31)^2 = 1369 - 144 = 1225$$

Step5: Take square root and solve for b

$$b+31 = \pm\sqrt{1225} = \pm35$$
Solve for positive $b$ (matches options):
$$b = 35 - 31 = 4$$
(The negative solution $b=-66$ is not among the options)

Answer:

4