If z_1 and z_2 are any two complex numbers,th | vedclass

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(A) Let $z_1 + \sqrt{z_1^2 - z_2^2} = u$ and $z_1 - \sqrt{z_1^2 - z_2^2} = v$. Then $u + v = 2z_1$ and $uv = z_1^2 - (z_1^2 - z_2^2) = z_2^2$. We want

Vedclass · Accepted Answer

$|z_1 + z_2| + |z_1 - z_2|$

Vedclass · Answer

(A) Let $z_1 + \sqrt{z_1^2 - z_2^2} = u$ and $z_1 - \sqrt{z_1^2 - z_2^2} = v$. Then $u + v = 2z_1$ and $uv = z_1^2 - (z_1^2 - z_2^2) = z_2^2$. We want to find $|u| + |v|$. Note that $(|u| + |v|)^2 = |u|^2 + |v|^2 + 2|uv| = u\bar{u} + v\bar{v} + 2|z_2^2|$. Alternatively,using the property $|u+v| + |u-v| = |u+v| + |u-v|$,we know that for any complex numbers $u, v$,$|u+v| + |u-v| = |z_1 + \sqrt{z_1^2 - z_2^2} + z_1 - \sqrt{z_1^2 - z_2^2}| + |z_1 + \sqrt{z_1^2 - z_2^2} - (z_1 - \sqrt{z_1^2 - z_2^2})| = |2z_1| + |2\sqrt{z_1^2 - z_2^2}|$. Testing with $z_1 = 5, z_2 = 3$: $|5 + \sqrt{25-9}| + |5 - \sqrt{25-9}| = |5+4| + |5-4| = 9 + 1 = 10$. Checking options: $|z_1+z_2| + |z_1-z_2| = |5+3| + |5-3| = 8 + 2 = 10$. Thus,the correct expression is $|z_1 + z_2| + |z_1 - z_2|$.

If $z_1$ and $z_2$ are any two complex numbers,then $|z_1 + \sqrt{z_1^2 - z_2^2}| + |z_1 - \sqrt{z_1^2 - z_2^2}|$ is equal to

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