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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
 #1
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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:

\[8a^2 + 2a - 5 = 0.\]

Using the quadratic formula:

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

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:

\[4a^2 + 2a - 5 = 0.\]

Using the quadratic formula again:

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

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

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