Prove that the points, represented by complex numbers ` (5+8i),(13+20i),(19+29i)` are collinear.

Answer

To prove that the points represented by the complex numbers \( z_1 = 5 + 8i \), \( z_2 = 13 + 20i \), and \( z_3 = 19 + 29i \) are collinear, we will use the property of collinearity in the context of complex numbers. The property states that three points \( z_1, z_2, z_3 \) are collinear if:

\[ |z_1 - z_2| + |z_2 - z_3| = |z_1 - z_3| \]

### Step 1: Calculate \( |z_1 - z_2| \)

First, we find \( z_1 - z_2 \):

\[ z_1 - z_2 = (5 + 8i) - (13 + 20i) = 5 - 13 + (8 - 20)i = -8 - 12i \]

Now, we calculate the modulus:

\[ |z_1 - z_2| = |-8 - 12i| = \sqrt{(-8)^2 + (-12)^2} = \sqrt{64 + 144} = \sqrt{208} = 4\sqrt{13} \]

### Step 2: Calculate \( |z_2 - z_3| \)

Next, we find \( z_2 - z_3 \):

\[ z_2 - z_3 = (13 + 20i) - (19 + 29i) = 13 - 19 + (20 - 29)i = -6 - 9i \]

Now, we calculate the modulus:

\[ |z_2 - z_3| = |-6 - 9i| = \sqrt{(-6)^2 + (-9)^2} = \sqrt{36 + 81} = \sqrt{117} = 3\sqrt{13} \]

### Step 3: Calculate \( |z_1 - z_3| \)

Now, we find \( z_1 - z_3 \):

\[ z_1 - z_3 = (5 + 8i) - (19 + 29i) = 5 - 19 + (8 - 29)i = -14 - 21i \]

Now, we calculate the modulus:

\[ |z_1 - z_3| = |-14 - 21i| = \sqrt{(-14)^2 + (-21)^2} = \sqrt{196 + 441} = \sqrt{637} = 7\sqrt{13} \]

### Step 4: Verify the collinearity condition

Now we check if the collinearity condition holds:

\[ |z_1 - z_2| + |z_2 - z_3| = 4\sqrt{13} + 3\sqrt{13} = 7\sqrt{13} \]

And we have:

\[ |z_1 - z_3| = 7\sqrt{13} \]

Since:

\[ |z_1 - z_2| + |z_2 - z_3| = |z_1 - z_3| \]

This confirms that the points are collinear.

### Conclusion

Thus, the points represented by the complex numbers \( (5 + 8i) \), \( (13 + 20i) \), and \( (19 + 29i) \) are collinear.

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