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Coordinates of a Point in Space

A particle st...

Question

A particle starts moving from a point P on the curve | z - 3 - 4i| = 5, where |z| is maximum. From P it moves through an angle $t a n^{- 1} (\frac{3}{4})$ in in anticlockwise direction and reaches the point Q. From Q it comes down parallel to imaginary axis by 2 units and reaches the point R. Then

A

R = (3, 9)

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B

R = (3, 7)

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C

Area of

Δ

PQR = 3 sq. Units

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D

Δ

PQR is right angled

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Solution

The correct options are
B $R = (3, 7)$
C Area of $Δ$ PQR = 3 sq. Units
|z - 3 - 4i| = 5 represent a circle $(x - 3)^{2} + (y - 4)^{2} = 25$ , which passes through O, P is the other end of diameter through 'O'
$∴$ P = (6, 8)
Q is vertically above C(3, 4)
$∴$ Q = (3, 9)
$\Rightarrow$ R = (3, 7)

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Q. A particle starts to travel from a pt. 'P' on curve

C_{1} : | z - 3 - 4 i | = 5

| z | \to

max. From 'P', the particle moves through an

∠

t a n^{- 1} (\frac{3}{4})

in Anti-CW direction on

C_{2}

and reaches Q. From Q, it comes down parallel to

I_{m}

-axis by 2 units & reaches R.
Find the co-ordinates of R.

P

starts from the point

z_{0} = 1 + 2 i

i = \sqrt{- 1}

, It moves first horizontally away from origin by

5

units and then vertically away from origin by

3

units to each a point

z_{1}

z_{1}

\sqrt{2}

units in the direction of vector

^i +^j

and then it moves through an angle in anticlockwise direction on a circle with centre at origin. To reach a point

z_{2}

z_{2}

P

starts from the point

z_{0} = 1 + 2 i

i = \sqrt{- 1}

. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point

z_{1}

z_{1}

the particle moves

\sqrt{2}

units in the direction of the vector

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and then it moves through an angle

\frac{π}{2}

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5

units and then vertically away from origin by

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z_{1}

the particle moves

\sqrt{2}

units away from origin in the direction of

x = y

and then it moves through an angle

π / 2

in anticlockwise direction on a circle with centre at origin to reach a point

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z_{0} = 1 + 2 i

i = \sqrt{- 1}

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the particle moves

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