Question

Determine the modulus of each of the complex numbers in the matching pairs. List these moduli in order from smallest to largest. Use the letters of the matching pair in your listing.

Suppose m = 2 + 6i, and |m + n| = 3√10 , where n is a complex number.

a. What is the minimum value of the modulus of n?

√10

Provide one example of the complex number, n.



A = J → 2 + 6i

B = F →75

C = H → -3 - i

D = I → 1 + i

E = G → 5 + 5i

these are my matching pairs

Answer 1

Let n be x+yi

So

• √(a²+b²+2²+6²)=3√10
• √a²+b²+40=3√10
• a²+b²+40=90
• a²+b²=50

Minimum value is 5+5i as 5²+5²=50

Answer 2

the modulus of a complex number z = a + bi is:

Izl= √(a²+b²)

The fact that n is complex does not mean that n doesn't has a real part, so we must write our numbers as:

m = 2 + 6i

n = a + bi

Im + nl = 3√10

√(a² + b²+ 2²+ 6²)= 3√10

√(a^2 + b^2 + 40) = 3√10

squaring both side

a²+b²+40 = 3^2*10 = 9*10 =90

a²+b²= 90 - 40

a²+b²=50

So,

|n|=√(a^2 + b^2) = √50

The modulus of n must be equal to the square root of 50.

now

values a and b such

a^2 + b^2 = 50.

for example, a = 5 and b = 5

5²+5²=25+25= 50

Then a possible value for n is:

n = 5+5i