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Answer :
To find the value of [tex]\( b \)[/tex] such that the point [tex]\( 25 + bi \)[/tex] is 37 units away from the point [tex]\( 13 - 31i \)[/tex], we can use the distance formula for complex numbers. The distance [tex]\( d \)[/tex] between two complex numbers [tex]\( z_1 = x_1 + y_1i \)[/tex] and [tex]\( z_2 = x_2 + y_2i \)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
In this problem, we have:
- [tex]\( z_1 = 25 + bi \)[/tex]
- [tex]\( z_2 = 13 - 31i \)[/tex]
Plugging these values into the distance formula gives us:
[tex]\[
\sqrt{(13 - 25)^2 + ((-31) - b)^2} = 37
\][/tex]
Now let's calculate each part:
1. The real part difference: [tex]\( 13 - 25 = -12 \)[/tex].
2. Square this difference: [tex]\((-12)^2 = 144\)[/tex].
Next, consider the imaginary parts:
1. The imaginary part difference: [tex]\((-31) - b = -31 - b \)[/tex].
2. Square this difference: [tex]\((-31 - b)^2\)[/tex].
The equation becomes:
[tex]\[
\sqrt{144 + (-31 - b)^2} = 37
\][/tex]
To remove the square root, square both sides of the equation:
[tex]\[
144 + (-31 - b)^2 = 37^2
\][/tex]
Calculate [tex]\( 37^2 \)[/tex]:
[tex]\[
37^2 = 1369
\][/tex]
Thus, the equation is:
[tex]\[
144 + (-31 - b)^2 = 1369
\][/tex]
Subtract 144 from both sides:
[tex]\[
(-31 - b)^2 = 1369 - 144
\][/tex]
Calculate the right side:
[tex]\[
1369 - 144 = 1225
\][/tex]
So we have:
[tex]\[
(-31 - b)^2 = 1225
\][/tex]
Take the square root of both sides:
[tex]\[
-31 - b = \pm \sqrt{1225}
\][/tex]
Calculate the square root:
[tex]\[
\sqrt{1225} = 35
\][/tex]
Thus, we have two possible equations:
1. [tex]\(-31 - b = 35\)[/tex]
2. [tex]\(-31 - b = -35\)[/tex]
Solve each equation for [tex]\( b \)[/tex]:
1. For [tex]\(-31 - b = 35\)[/tex]:
[tex]\[
-31 - b = 35 \\
-b = 35 + 31 \\
-b = 66 \\
b = -66
\][/tex]
2. For [tex]\(-31 - b = -35\)[/tex]:
[tex]\[
-31 - b = -35 \\
-b = -35 + 31 \\
-b = -4 \\
b = 4
\][/tex]
The possible values of [tex]\( b \)[/tex] that satisfy the condition are [tex]\(-66\)[/tex] and [tex]\(4\)[/tex].
From the provided options, the value [tex]\( b = 4 \)[/tex] is present. Thus, the correct answer is:
4
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
In this problem, we have:
- [tex]\( z_1 = 25 + bi \)[/tex]
- [tex]\( z_2 = 13 - 31i \)[/tex]
Plugging these values into the distance formula gives us:
[tex]\[
\sqrt{(13 - 25)^2 + ((-31) - b)^2} = 37
\][/tex]
Now let's calculate each part:
1. The real part difference: [tex]\( 13 - 25 = -12 \)[/tex].
2. Square this difference: [tex]\((-12)^2 = 144\)[/tex].
Next, consider the imaginary parts:
1. The imaginary part difference: [tex]\((-31) - b = -31 - b \)[/tex].
2. Square this difference: [tex]\((-31 - b)^2\)[/tex].
The equation becomes:
[tex]\[
\sqrt{144 + (-31 - b)^2} = 37
\][/tex]
To remove the square root, square both sides of the equation:
[tex]\[
144 + (-31 - b)^2 = 37^2
\][/tex]
Calculate [tex]\( 37^2 \)[/tex]:
[tex]\[
37^2 = 1369
\][/tex]
Thus, the equation is:
[tex]\[
144 + (-31 - b)^2 = 1369
\][/tex]
Subtract 144 from both sides:
[tex]\[
(-31 - b)^2 = 1369 - 144
\][/tex]
Calculate the right side:
[tex]\[
1369 - 144 = 1225
\][/tex]
So we have:
[tex]\[
(-31 - b)^2 = 1225
\][/tex]
Take the square root of both sides:
[tex]\[
-31 - b = \pm \sqrt{1225}
\][/tex]
Calculate the square root:
[tex]\[
\sqrt{1225} = 35
\][/tex]
Thus, we have two possible equations:
1. [tex]\(-31 - b = 35\)[/tex]
2. [tex]\(-31 - b = -35\)[/tex]
Solve each equation for [tex]\( b \)[/tex]:
1. For [tex]\(-31 - b = 35\)[/tex]:
[tex]\[
-31 - b = 35 \\
-b = 35 + 31 \\
-b = 66 \\
b = -66
\][/tex]
2. For [tex]\(-31 - b = -35\)[/tex]:
[tex]\[
-31 - b = -35 \\
-b = -35 + 31 \\
-b = -4 \\
b = 4
\][/tex]
The possible values of [tex]\( b \)[/tex] that satisfy the condition are [tex]\(-66\)[/tex] and [tex]\(4\)[/tex].
From the provided options, the value [tex]\( b = 4 \)[/tex] is present. Thus, the correct answer is:
4
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