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# help

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Help I have trouble with complex numbers.

In the complex plane, the complex numbers $2+i$, $a$, $7-3i$, $b$ form the vertices of a square, in counterclockwise order.  Find the following quantities, in rectangular form.

b/a = ?

Aug 16, 2023

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The vertices of the square are given as complex numbers: $2+i$, $a$, $7-3i$, and $b$. Since these points form a square, the diagonals are perpendicular, and the midpoint of one diagonal is the center of the square. Let's use this information to solve for $$a$$ and $$b$$.

The midpoint of the diagonal connecting $2+i$ and $7-3i$ can be found as:

$\text{Midpoint} = \frac{(2+i) + (7-3i)}{2} = \frac{9 - 2i}{2} = \frac{9}{2} - i.$

This midpoint should be equal to the midpoint of the diagonal connecting $a$ and $b$:

$\frac{a + b}{2} = \frac{9}{2} - i.$

From this equation, we can solve for $$b$$:

$b = 9 - 2a - 2i.$

Now, since the vertices are arranged in counterclockwise order, $$2+i$$ and $$a$$ are adjacent vertices of the square. The side connecting these two vertices has the same length as the side connecting $$7-3i$$ and $$b$$. This gives us the following relationship:

$\|2+i - a\| = \|7-3i - b\|.$

Substituting the expressions for $$a$$ and $$b$$, we get:

$\|2+i - a\| = \|7-3i - (9 - 2a - 2i)\|.$

Simplify the expressions:

$\|2+i - a\| = \|2a - 2i\|.$

Since the lengths of the sides are equal, the magnitudes of the differences are equal:

$\|2+i - a\| = \|2a - 2i\|.$

Square both sides of the equation:

$(2+i - a)(\overline{2+i - a}) = (2a - 2i)(\overline{2a - 2i}).$

Expand both sides:

$(2+i - a)(2-i - \bar{a}) = (2a - 2i)(2a + 2i).$

Simplify further:

$(4 - a - ai + 2i - 2i + i^2 + \bar{a} - \bar{a}i + a\bar{a}) = (4a^2 - 4i^2).$

Since $$i^2 = -1$$:

$(4 - 2ai + a^2 + \bar{a} - a\bar{a}) = (4a^2 + 4).$

Rearrange the terms:

$(4 + 4a^2) - 2ai + a^2 + \bar{a} - a\bar{a} = 4a^2 + 4.$

Simplify and collect the real and imaginary terms:

$5 - 3a^2 - 2ai + \bar{a} - a\bar{a} = 0.$

Since $$a$$ is a real number, $$\bar{a} = a$$:

$5 - 3a^2 - 2ai + a - a^2 = 0.$

Combine like terms:

$5 - 4a^2 - 2ai + a = 0.$

Solve for $$a$$:

$5 - 4a^2 - ai = -a.$

$5 - 4a^2 = -ai + a.$

$5 - 4a^2 = a(1 - i).$

$a = \frac{5 - 4a^2}{1 - i}.$

Multiply the numerator and denominator by the conjugate of the denominator to rationalize it:

$a = \frac{(5 - 4a^2)(1 + i)}{(1 - i)(1 + i)}.$

$a = \frac{5 + 5i - 4a^2 - 4a^2i}{2}.$

$a = \frac{5 - 8a^2 + 5i - 4a^2i}{2}.$

Equating the real and imaginary parts separately:

$a = \frac{5 - 8a^2}{2} \quad \text{(real part)},$
$a = \frac{5 - 4a^2}{2} \quad \text{(imaginary part)}.$

Solve each equation for $$a$$:

$5 - 8a^2 = 2a.$
$5 - 4a^2 = 2a.$

For the first equation:

$8a^2 + 2a - 5 = 0.$

$a = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 8 \cdot (-5)}}{2 \cdot 8}.$

Simplify under the square root:

$a = \frac{-2 \pm \sqrt{4 + 160}}{16}.$

$a = \frac{-2 \pm \sqrt{164}}{16}.$

Since $$164$$ is not a perfect square, we'll leave it as is:

$a = \frac{-2 \pm \sqrt{41} \sqrt{4}}{16}.$

$a = \frac{-1 \pm \sqrt{41}}{8}.$

For the second equation:

$4a^2 + 2a - 5 = 0.$

$a = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 4 \cdot (-5)}}{2 \cdot 4}.$

Simplify under the square root:

$a = \frac{-2 \pm \sqrt{4 + 80}}{8}.$

$a = \frac{-2 \pm \sqrt{84}}{8}.$

Since $$84$$ is not a perfect square, we'll leave it as is:

$a = \frac{-2 \pm \sqrt{4} \sqrt{21}}{8}.$

$a = \frac{-1 \pm 2\sqrt{21}}{8}.$

So, the possible values for $$a$$ are:

$a = \frac{-1 + \sqrt{41}}{8}, \quad a = \frac{-1 - \sqrt{41}}{8}, \quad a = \frac{-1 + 2\sqrt{21}}{8}, \quad a = \frac{-1 - 2\sqrt{21}}{8}.$

Now, we have the expression $$b/a$$. Let's find the possible values of $$b$$ for each of the values of $$a$$:

For $$a = \frac{-1 + \sqrt{41}}{8}$$:

$b = 9 - 2a - 2i = 9 - 2\left(\frac{-1 + \sqrt{41}}{8}\right) - 2i.$

Simplify:

$b = \frac{9 + \sqrt{41}}{4} - 2i.$

For $$a = \frac{-1 - \sqrt{41}}{8}$$:

$b = 9 - 2a - 2i = 9 - 2\left(\frac{-1 - \sqrt{41}}{8}\right) - 2i.$

Simplify:

$b = \frac{9 - \sqrt{41}}{4} - 2i.$

For $$a = \frac{-1 + 2\sqrt{21}}{8}$$:

$b = 9 - 2a - 2i = 9 - 2\left(\frac{-1 + 2\sqrt{21}}{8}\right) - 2i.$

Simplify:

$b = \frac{9 + 2\sqrt{21}}{4} - 2i.$

For $$a = \frac{-1 - 2\sqrt{21}}{8}$$:

$b = 9 - 2a - 2i = 9 - 2\left(\frac{-1 - 2\sqrt{21}}{8}\right) - 2i.$

Simplify:

$b = \frac{9 - 2\sqrt{21}}{4} - 2i.$

So, the possible values of $$b/a$$ for each value of $$a$$ are:

For $$a = \frac{-1 + \sqrt{41}}{8}$$:

$\frac{b}{a} = \frac{\frac{9 + \sqrt{41}}{4} - 2i}{\frac{-1 + \sqrt{41}}{8}} = \frac{8(9 + \sqrt{41}) - 8(2i)}{4(-1 + \sqrt{41})}.$

Simplify:

$\frac{b}{a} = \frac{72 + 8\sqrt{41} - 16i}{-4 + 4\sqrt{41}}.$

For $$a = \frac{-1 - \sqrt{41}}{8}$$:

$\frac{b}{a} = \frac{\frac{9 - \sqrt{41}}{4} - 2i}{\frac{-1 - \sqrt{41}}{8}} = \frac{8(9 - \sqrt{41}) - 8(2i)}{4(-1 - \sqrt{41})}.$

Simplify:

$\frac{b}{a} = \frac{72 - 8\sqrt{41} - 16i}{-4 - 4\sqrt{41}}.$

For $$a = \frac{-1 + 2\sqrt{21}}{8}$$:

$\frac{b}{a} = \frac{\frac{9 + 2\sqrt{21}}{4} - 2i}{\frac{-1 + 2\sqrt{21}}{8}} = \frac{8(9 + 2\sqrt{21}) - 8(2i)}{4(-1 + 2\sqrt{21})}.$

Simplify:

$\frac{b}{a} = \frac{72 + 16\sqrt{21} - 16i}{-4 + 8\sqrt{21}}.$

For $$a = \frac{-1 - 2\sqrt{21}}{8}$$:

$\frac{b}{a} = \frac{\frac{9 - 2\sqrt{21}}{4} - 2i}{\frac{-1 - 2\sqrt{21}}{8}} = \frac{8(9 - 2\sqrt {21}) - 8(2i)}{4(-1 - 2\sqrt{21})}.$

Simplify:

$\frac{b}{a} = \frac{72 - 16\sqrt{21} - 16i}{-4 - 8\sqrt{21}}.$

In conclusion, the possible values of $$b/a$$ for the given values of $$a$$ are:

$\frac{b}{a} = \frac{72 + 8\sqrt{41} - 16i}{-4 + 4\sqrt{41}}, \quad \frac{b}{a} = \frac{72 - 8\sqrt{41} - 16i}{-4 - 4\sqrt{41}}, \quad \frac{b}{a} = \frac{72 + 16\sqrt{21} - 16i}{-4 + 8\sqrt{21}}, \quad \frac{b}{a} = \frac{72 - 16\sqrt{21} - 16i}{-4 - 8\sqrt{21}}.$

Aug 16, 2023