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|z1z_1|=\sqrt{1717},|z2z_2|=22\sqrt{22},|3z1+z23z_1+z_2|=\sqrt{221221},P=∣z2+z1−z1z2∣P=|z_2+z_1-z_1z_2|. Find exact value of PP

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Full solution

Q. |z1z_1|=\sqrt{1717},|z2z_2|=22\sqrt{22},|3z1+z23z_1+z_2|=\sqrt{221221},P=∣z2+z1−z1z2∣P=|z_2+z_1-z_1z_2|. Find exact value of PP
  1. Calculate Magnitude of z1∗z2z_1 * z_2: We know that ∣z1∣=17|z_1| = \sqrt{17} and ∣z2∣=22|z_2| = 2\sqrt{2}. To find PP, we need to calculate ∣z2+z1−z1∗z2∣|z_2 + z_1 - z_1 * z_2|.
  2. Find Bounds for ∣z2+z1∣|z_2 + z_1|: First, let's find the magnitude of z1∗z2z_1 * z_2 using the property ∣z1∗z2∣=∣z1∣∗∣z2∣|z_1 * z_2| = |z_1| * |z_2|.∣z1∗z2∣=17∗22=(17∗4∗2)=(17∗8)=136.|z_1 * z_2| = \sqrt{17} * 2\sqrt{2} = \sqrt{(17 * 4 * 2)} = \sqrt{(17 * 8)} = \sqrt{136}.
  3. Limitation in Finding Exact Value: Now, we don't have the exact values of z1z_1 and z2z_2, but we can use the triangle inequality to find the bounds for ∣z2+z1∣|z_2 + z_1|. The triangle inequality states that ∣z2+z1∣≤∣z2∣+∣z1∣|z_2 + z_1| \leq |z_2| + |z_1|. So, ∣z2+z1∣≤22+17∣|z_2 + z_1| \leq 2\sqrt{2} + \sqrt{17}|.
  4. Limitation in Finding Exact Value: Now, we don't have the exact values of z1z_1 and z2z_2, but we can use the triangle inequality to find the bounds for ∣z2+z1∣|z_2 + z_1|. The triangle inequality states that ∣z2+z1∣≤∣z2∣+∣z1∣|z_2 + z_1| \leq |z_2| + |z_1|. So, ∣z2+z1∣≤22+17|z_2 + z_1| \leq 2\sqrt{2} + \sqrt{17}. However, we need the exact value of ∣z2+z1∣|z_2 + z_1|, not just an inequality. We can't find the exact value without more information about z1z_1 and z2z_2.

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