Ifarg z-a - - -4- where a belongs to R- then the locus of z belongs to c is a

Finding the value:The Given arg (z a) = π/4Let z=x+iy⇒ arg(x+iy a) = π/40ex0ex⇒ arg((x a)+iy) = π/40ex0ex⇒tan^ 1(y/(x a))= π/40ex0ex⇒y/ (x a)= tanπ/40ex0ex ...

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Byju's Answer

Standard XII

Mathematics

Addition of Vectors

Ifarg z-a = π...

Question

If $a r g (z - a) = π / 4$ , where a belongs to $R$ , then the locus of $z$ belongs to $c$ is a

Hyperbola

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Parabola

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Ellipse

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Straight line

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Solution

The correct option is D
Straight line

Finding the value:
The Given $a r g (z - a) = \frac{π}{4}$
Let $z = x + i y$
$\Rightarrow a r g (x + i y - a) = \frac{π}{4} \Rightarrow a r g ((x - a) + i y) = \frac{π}{4} \Rightarrow \tan^{- 1} (\frac{y}{(x - a)}) = \frac{π}{4} \Rightarrow \frac{y}{(x - a)} = \tan \frac{π}{4} \Rightarrow \frac{y}{(x - a)} = 1 \Rightarrow y = x - a$
Thus, $x - y = a$ represents a straight line.
Hence, the correct option is D.

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Similar questions

Q. If

z = \cos \frac{π}{4} + i \sin \frac{π}{6}

, then
(a)

|z| = 1, a r g (z) = \frac{π}{4}

(b)

|z| = 1, a r g (z) = \frac{π}{6}

(c)

|z| = \frac{\sqrt{3}}{2}, a r g (z) = \frac{5 π}{24}

(d)

|z| = \frac{\sqrt{3}}{2}, a r g (z) = \tan^{- 1} \frac{1}{\sqrt{2}}

Q. If

a r g (z - α) = \frac{π}{4}

, where

α \in R

, then the locus of

z \in Z

is a

Q. The locus of the complex number

z = x + i y

where

i = \sqrt{- 1}

satisfying relation

a r g (z - a) = \frac{π}{4}

where

a \in R

Q. For positive constant

r,

let

M

be the set of complex numbers

z

which satisfy

| z - 4 - 3 i | = r .

Then which of the following statements is (are) CORRECT?

Q. If

| z - 2 - i | = | z | | sin (\frac{π}{4} - a r g z) |

, then locus of

z

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Ifarg(z-a)=π/4, where a belongs to R, then the locus of z belongs to c is a

The correct option is D Straight lineFinding the value:The Given arg(z-a)=π4Let z=x+iy⇒arg(x+iy-a)=π4⇒arg((x-a)+iy)=π4⇒tan-1y(x-a)=π4⇒y(x-a)=tanπ4⇒y(x-a)=1⇒y=x-aThus,x-y=a represents a straight line.Hence, the correct option is D.

If $a r g (z - a) = π / 4$ , where a belongs to $R$ , then the locus of $z$ belongs to $c$ is a