The complex number having least positive argument and satisfying the inequality $| z - 5 i | \leq 3$ is

\frac{12}{5} + \frac{16}{5} i

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\frac{12}{5} + \frac{12}{5} i

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\frac{16}{5} + \frac{12}{5} i

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\frac{3}{5} + \frac{4}{5} i

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Solution

The correct option is A $\frac{12}{5} + \frac{16}{5} i$
$| z - 5 i | \leq 3$
The above inequality represents all the points which lie inside the circle or on the circle having centre $(0, 5)$ with radius of $3.$
So, for least argument, tangent is drawn to the circle at $A$ passsing through the origin.
$\to$ Complex number with least argument is A

$O A = 4$
So the coordinates of $A$ will be $(4 cos θ, 4 sin θ)$
From $△ C A O,$
$sin θ = \frac{4}{5} & cos θ = \frac{3}{5}$
So required complex number is $\frac{12}{5} + \frac{16}{5} i$

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Similar questions

Verify the following :

(i) $\frac{3}{7} \times (\frac{5}{6} + \frac{12}{13}) = (\frac{3}{7} \times \frac{5}{6}) + (\frac{3}{7} \times \frac{12}{13})$
(ii) $\frac{- 15}{4} \times (\frac{3}{7} + \frac{- 12}{5}) = (\frac{- 15}{4} \times \frac{3}{7}) + (\frac{- 15}{4} \times \frac{- 12}{5})$
(iii) $(\frac{- 8}{3} + \frac{- 13}{12}) \times \frac{5}{6} = (\frac{- 8}{3} \times \frac{5}{6}) + (\frac{- 13}{12} \times \frac{5}{6})$
(iv) $\frac{- 16}{7} \times (\frac{- 8}{9} + \frac{- 7}{6}) = (\frac{- 16}{7} \times \frac{- 8}{9}) + (\frac{- 16}{7} \times \frac{- 7}{6})$

Find the complex number z satisfying the equations $∣ ∣ \frac{z - 12}{z - 8 i} ∣ ∣$ = $\frac{5}{3}$ , $∣ ∣ \frac{z - 4}{z - 8} ∣ ∣$ =1

Q. The sum of the series

\frac{3}{4 \cdot 8} - \frac{3 \cdot 5}{4 \cdot 8 \cdot 12} + \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12 \cdot 16} - . . . . . .

Find the complex numbers $z$ which simultaneously satisfy the equations

$∣ ∣ ∣ \frac{z - 12}{z - 8 i} ∣ ∣ ∣ = \frac{5}{3}$ and $∣ ∣ ∣ \frac{z - 4}{z - 8} ∣ ∣ ∣ = 1$ .

Q. Find the complex numbers z which simultaneously satisfy the equation

∣ ∣ ∣ \frac{z - 12}{z - 8 i} ∣ ∣ ∣ = \frac{5}{3}

and

∣ ∣ ∣ \frac{z - 4}{z - 8} ∣ ∣ ∣ = 1

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The complex number having least positive argument and satisfying the inequality |z−5i|≤3 is

The complex number having least positive argument and satisfying the inequality $| z - 5 i | \leq 3$ is