If 3- z 1-2- z 12- z 2-3- z 2- k- then points A z 1- B z 2- C 3-0 and D 2-0 taken in clockwise sense willA- Lie on a circle only for k-0B- Lie on a circle only for k-0C- Lie on a circle - k - RD- Be vertices of a square - k -0-1

The correct option is A arg3 z_1/2 z_1 +arg2 z_2/3 z_2 = arg3 z_1/2 z_12 z_2/3 z_2 If k gt; 0, its argument will be zero So, θ_1 amp; θ_2 are equal in m ...

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Byju's Answer

Standard XII

Mathematics

Properties of Modulus

If 3- z 1/2- ...

Question

If $⎧ ⎩ \frac{3 - z_{1}}{2 - z_{1}} ⎫ ⎭ ⎧ ⎩ \frac{2 - z_{2}}{3 - z_{2}} ⎫ ⎭$ = k, then points $A (z_{1})$ , $B (z_{2})$ , $C (3, 0)$ and $D (2, 0)$ (taken in clockwise sense) will

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Solution

The correct option is A

arg $⎧ ⎩ \frac{3 - z_{1}}{2 - z_{1}} ⎫ ⎭ + a r g ⎧ ⎩ \frac{2 - z_{2}}{3 - z_{2}} ⎫ ⎭$

= $a r g ⎧ ⎩ \frac{3 - z_{1}}{2 - z_{1}} ⎫ ⎭ ⎧ ⎩ \frac{2 - z_{2}}{3 - z_{2}} ⎫ ⎭$

If k > 0, its argument will be zero

So, $θ_{1}$ & $θ_{2}$ are equal in magnitude but opposite sign.

So DC subtends equal qngle at A & B. So, points are concyclic if k > 0

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Similar questions

Q. If

(\frac{3 - z_{1}}{2 - z_{1}}) (\frac{2 - z_{2}}{3 - z_{2}}) = k;

then points

A (z 1), B (z 2), C (2, 0)

and

D (3, 0)

will

Q. If

(\frac{3 - z_{1}}{2 - z_{1}}) (\frac{2 - z_{2}}{3 - z_{2}}) = k (k > 0)

Then prove that the points

A (z_{1}), B (z_{2}), C (3)

and

D (2)

(taken in clockwise sense) are concyclic.

Q. Four distinct points (2k, 3k), (1, 0) (0, 1) and (0, 0) lie on a circle for

Q. Let

a

be complex number such that

| a | < 1

and

z_{1}, z_{2}, z_{3}, . . .

be the vertices of a polygon such that

z_{k} = 1 + a + a^{2} + . . + a^{k - 1}

for all

k = 1, 2. 3, . . .

Then

z_{1}, z_{2}, . . .

lie within the circle

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If ⎧⎩3−z12−z1⎫⎭⎧⎩2−z23−z2⎫⎭ = k, then points A(z1), B(z2), C(3,0) and D(2,0) (taken in clockwise sense) will

The correct option is A arg⎧⎩3−z12−z1⎫⎭+arg⎧⎩2−z23−z2⎫⎭ = arg⎧⎩3−z12−z1⎫⎭⎧⎩2−z23−z2⎫⎭ If k > 0, its argument will be zero So, θ1 & θ2 are equal in magnitude but opposite sign. So DC subtends equal qngle at A & B. So, points are concyclic if k > 0

If $⎧ ⎩ \frac{3 - z_{1}}{2 - z_{1}} ⎫ ⎭ ⎧ ⎩ \frac{2 - z_{2}}{3 - z_{2}} ⎫ ⎭$ = k, then points $A (z_{1})$ , $B (z_{2})$ , $C (3, 0)$ and $D (2, 0)$ (taken in clockwise sense) will