The number of solutions of equation -z- - -z - 1- - -z - 2- - -z - 3- - 4-and-z z- - z -z- - 5-4 where z is a complex number -021Infinite

The correct option is A 0z z̅ z z̅ = 5/4 z = x + iy ⇒ x^2 + y^2 2x 5/4 = 0.....(i) |z| + |z – 1| + |z – 2| + |z – 3| = 4 ....(ii) ∵ |z| + |z 1| + |z ...

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Standard XII

Mathematics

Conjugate of a Complex Number

The number of...

Question

The number of solutions of equation $| z | + | z - 1 | + | z - 2 | + | z - 3 | = 4 a n d z ¯ z - z - ¯ z = \frac{5}{4}$ where z is a complex number ?

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Solution

$z ¯ z - z - ¯ z = \frac{5}{4} z = x + i y$
$\Rightarrow x^{2} + y^{2} - 2 x - \frac{5}{4} = 0$ .....(i)
$| z | + | z - 1 | + | z - 2 | + | z - 3 | = 4$ ....(ii)
$∵ | z | + | z - 1 | + | z - 2 | + | z - 3 | \geq 4$
$∴$ No solution

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The number of solutions of equation |z|+|z−1|+|z−2|+|z−3|=4 and z¯z−z−¯z=54 where z is a complex number ?

z¯z−z−¯z=54z= x+iy ⇒x2+y2−2x−54=0.....(i) |z|+|z–1|+|z–2|+|z–3|=4 ....(ii) ∵|z|+|z−1|+|z−2|+|z−3|≥4 ∴ No solution

The number of solutions of equation $| z | + | z - 1 | + | z - 2 | + | z - 3 | = 4 a n d z ¯ z - z - ¯ z = \frac{5}{4}$ where z is a complex number ?

$z ¯ z - z - ¯ z = \frac{5}{4} z = x + i y$
$\Rightarrow x^{2} + y^{2} - 2 x - \frac{5}{4} = 0$ .....(i)
$| z | + | z - 1 | + | z - 2 | + | z - 3 | = 4$ ....(ii)
$∵ | z | + | z - 1 | + | z - 2 | + | z - 3 | \geq 4$
$∴$ No solution