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Complex Numbers question

2022 · 27 Jul · Shift 1 · Q26
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  5. /2022 · 27 Jul · Shift 1 · Q26

Complex Numbers question

2022 · 27 Jul · Shift 1 · Q26

JEE MainMathematicsComplex NumbersMCQ+4 / −1
Let the minimum value v0v_{0}v0​ of v=∣z∣2+∣z−3∣2+∣z−6i∣2,z∈Cv=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}v=∣z∣2+∣z−3∣2+∣z−6i∣2,z∈C is attained at z=z0{ }{z}=z_{0}z=z0​. Then ∣2z02−zˉ03+3∣2+v02\left|2 z_{0}^{2}-\bar{z}_{0}^{3}+3\right|^{2}+v_{0}^{2}​2z02​−zˉ03​+3​2+v02​ is equal to :
  1. A
    1000
  2. B
    1024
  3. C
    1105
  4. D
    1196
View written solutionFree

Correct answer: A

  1. Write zzz in Cartesian form

Let z=x+iy,z=x+iy,z=x+iy, so that ∣z∣2=x2+y2,|z|^2=x^2+y^2,∣z∣2=x2+y2, ∣z−3∣2=(x−3)2+y2,|z-3|^2=(x-3)^2+y^2,∣z−3∣2=(x−3)2+y2, ∣z−6i∣2=x2+(y−6)2.|z-6i|^2=x^2+(y-6)^2.∣z−6i∣2=x2+(y−6)2.

Hence, v=∣z∣2+∣z−3∣2+∣z−6i∣2v=|z|^2+|z-3|^2+|z-6i|^2v=∣z∣2+∣z−3∣2+∣z−6i∣2 becomes v=(x2+y2)+((x−3)2+y2)+(x2+(y−6)2).v=(x^2+y^2)+((x-3)^2+y^2)+(x^2+(y-6)^2).v=(x2+y2)+((x−3)2+y2)+(x2+(y−6)2).

  1. Expand and simplify vvv

v=x2+y2+x2−6x+9+y2+x2+y2−12y+36v=x^2+y^2+x^2-6x+9+y^2+x^2+y^2-12y+36v=x2+y2+x2−6x+9+y2+x2+y2−12y+36 =3x2+3y2−6x−12y+45.=3x^2+3y^2-6x-12y+45.=3x2+3y2−6x−12y+45.

Now complete squares: v=3(x2−2x)+3(y2−4y)+45v=3(x^2-2x)+3(y^2-4y)+45v=3(x2−2x)+3(y2−4y)+45 =3[(x−1)2−1]+3[(y−2)2−4]+45=3[(x-1)^2-1]+3[(y-2)^2-4]+45=3[(x−1)2−1]+3[(y−2)2−4]+45 =3(x−1)2+3(y−2)2+30.=3(x-1)^2+3(y-2)^2+30.=3(x−1)2+3(y−2)2+30.

Therefore, the minimum value is v0=30,v_0=30,v0​=30, attained at

\quad y=2.$$ So, $$z_0=1+2i.$$ 3. **Compute $\left|2z_0^2-\bar z_0^3+3\right|^2$** We have $$z_0=1+2i, \qquad \bar z_0=1-2i.$$ First, $$z_0^2=(1+2i)^2=1+4i-4=-3+4i,$$ so $$2z_0^2=-6+8i.$$ Next, $$\bar z_0^2=(1-2i)^2=1-4i-4=-3-4i,$$ $$\bar z_0^3=(1-2i)(-3-4i).$$ Multiply: $$(-3-4i)(1-2i)=-3+6i-4i+8i^2=-3+2i-8=-11+2i.$$ Thus, $$\bar z_0^3=-11+2i.$$ Now, $$2z_0^2-\bar z_0^3+3=(-6+8i)-(-11+2i)+3$$ $$=-6+8i+11-2i+3=8+6i.$$ Therefore, $$\left|2z_0^2-\bar z_0^3+3\right|^2=|8+6i|^2=8^2+6^2=100.$$ 4. **Compute the required expression** $$v_0^2=30^2=900.$$ So, $$\left|2z_0^2-\bar z_0^3+3\right|^2+v_0^2=100+900=1000.$$ 5. **Compare with stored answer** Derived answer is **1000**, which matches option **A** and agrees with the stored correct answer.
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