SOLUTION: How do you solve the problem (1-4i)/(2+3i) i is an imaginary number (the square root of -1) Also, how would you find the equation of a circle with a diameter AB, given that

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: How do you solve the problem (1-4i)/(2+3i) i is an imaginary number (the square root of -1) Also, how would you find the equation of a circle with a diameter AB, given that       Log On


   

Question 581638: How do you solve the problem (1-4i)/(2+3i)
i is an imaginary number (the square root of -1)
Also, how would you find the equation of a circle with a diameter AB, given that the coordinates of a and B are (-6,1) and (4,-5). What is the answer in standard form
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
%281-4i%29%2F%282%2B3i%29

The denominator is 2+3i.  Its conjugate is 2-3i 

(change the sign of the term containing i)

Multiply by the unit fraction formed by putting the con=jugate
of the denominator over itself: red%28%282-3i%29%2F%282-3i%29%29

%281-4i%29%2F%282%2B3i%29·red%28%282-3i%29%2F%282-3i%29%29

%28%281-4i%29%282-3i%29%29%2F%28%282%2B3i%29%282-3i%29%29

Multiply (FOIL) out the top and bottom:

%282-3i-8i%2B12i%5E2%29%2F%284-6i%2B6i-9i%5E2%29

Combine the like terms. (the two middle terms in the bottom cancel)  

%282-11i%2B12i%5E2%29%2F%284-9i%5E2%29

Substitute (-1) for i²

%282-11i%2B12%28-1%29%29%2F%284-9%28-1%29%29

Simplify

%282-11i-12%29%2F%284%2B9%29

Combine like terms:

%28-10-11i%29%2F13

Divide the -10 and the -11 each by 13

%28-10%29%2F13+-+expr%2811%2F13%29i

-10%2F13+-+expr%2811%2F13%29i

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Also, how would you find the equation of a circle with a diameter AB, given that the coordinates of a and B are (-6,1) and (4,-5). What is the answer in standard form:

We need the the radius r and the center (h,k) and then we can 
substitute thes is the equation for a circle:

(x - h)² + (y - k)² = r²

Let's graph the points and draw the diameter:



We will use the distance formula to find the diameter, which we will take
on half of for the radius. The distance formula is:

d = sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29

d = sqrt%28%28%284%29-%28-6%29%29%5E2%2B%28%28-5%29-%281%29%29%5E2%29%29 = sqrt%28%284%2B6%29%5E2%2B%28-5-1%29%5E2%29%29 = sqrt%2810%5E2%2B%28-6%29%5E2%29%29 = sqrt%28100%2B36%29 = sqrt%28136%29

d = sqrt%284%2A34%29 = sqrt%284%29sqrt%2834%29 = 2sqrt%2834%29

So the diameter is 2sqrt%2834%29 and the radius is one-half of that 
diameter which is sqrt%2834%29
 
The midpoint of any diameter is the center.  So we use the midpoint formula:

Midpoint =  = 
 = %28matrix%281%2C3%2C++++++%28-6%2B4%29%2F2%2C+++%22%2C%22%2C+%281-5%29%2F2%29%29 = %28matrix%281%2C3%2C++++++%28-2%29%2F2%2C+++%22%2C%22%2C+%28-4%29%2F2%29%29 = %28matrix%281%2C3%2C++++++-1%2C+++%22%2C%22%2C+-2%29%29

So the center is (h,k) = C(-1,-2) and the radius is r = sqrt%2834%29.

Substituting in

(x - h)² + (y - k)² = r²


(x - (-1))² + (y - (-2))² = (sqrt%2834%29)²

(x + 1)² + (y + 2)² = 34




Edwin