There exist two complex numbers c, say c1 and c2, so that 2 + 2i, 5 + i, and c form the vertices of an equilateral triangle. Find the product c1 c2.
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To find the product c1 * c2, we need to first find the complex numbers c1 and c2.
Given that the vertices of the equilateral triangle are 2 + 2i, 5 + i, and c, we can use some properties of equilateral triangles to solve this problem.
For an equilateral triangle, the distance between any two vertices is equal. So, we can set up equations to find the distances between the given vertices:
Distance between 2 + 2i and 5 + i:
Let's call this distance d1.
d1 = |(5 + i) - (2 + 2i)|
= |5 + i - 2 - 2i|
= |3 - i|
= √(3^2 + (-1)^2)
= √10
Distance between 2 + 2i and c:
Let's call this distance d2.
d2 = |c - (2 + 2i)|
= |c - 2 - 2i|
= √[(Re(c) - 2)^2 + (Im(c) - 2)^2]
Since the distance between any two vertices of an equilateral triangle is the same, we can set up an equation using d1 and d2:
√10 = √[(Re(c) - 2)^2 + (Im(c) - 2)^2]
Square both sides to eliminate the square root:
10 = (Re(c) - 2)^2 + (Im(c) - 2)^2
Expand and simplify:
10 = Re(c)^2 - 4Re(c) + 4 + Im(c)^2 - 4Im(c) + 4
10 = Re(c)^2 + Im(c)^2 - 4Re(c) - 4Im(c) + 8
10 = Re(c^2) - 4Re(c) + 8
Now, let's rewrite the complex number c as c = x + yi, where x is the real part and y is the imaginary part.
Substitute c = x + yi into the above equation:
10 = (x + yi)^2 - 4(x + yi) + 8
10 = (x^2 + 2xyi - y^2) - 4x - 4yi + 8
10 = (x^2 - y^2 - 4x + 8) + (2xy - 4y)i
Now, we can equate the real and imaginary parts on both sides:
Real part:
x^2 - y^2 - 4x + 8 = 10
Imaginary part:
2xy - 4y = 0
Let's solve the imaginary part equation:
2xy - 4y = 0
2y(x - 2) = 0
Either y = 0 or x - 2 = 0.
If y = 0, then c is purely real, which doesn't satisfy the given conditions of forming an equilateral triangle.
If x - 2 = 0, then x = 2, and substituting x = 2 into the real part equation:
2^2 - y^2 - 4(2) + 8 = 10
4 - y^2 - 8 + 8 = 10
4 - y^2 = 10 - 4
-y^2 = 6 - 4
-y^2 = 2
y^2 = -2
This equation has no real solutions for y, but it does have complex solutions.
Therefore, there is no valid complex number c that forms an equilateral triangle with the given vertices.
Since there is no valid c, there are no valid c1 and c2, and therefore the product c1 * c2 is undefined.