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Complex Numbers question

2023 · 1 Feb · Shift 1 · Q28
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  5. /2023 · 1 Feb · Shift 1 · Q28

Complex Numbers question

2023 · 1 Feb · Shift 1 · Q28

JEE MainMathematicsComplex NumbersMCQ+4 / −1
If the center and radius of the circle ∣z−2z−3∣=2\left| {{{z - 2} \over {z - 3}}} \right| = 2​z−3z−2​​=2 are respectively (α,β)(\alpha,\beta)(α,β) and γ\gammaγ, then 3(α+β+γ)3(\alpha+\beta+\gamma)3(α+β+γ) is equal to :
  1. A
    12
  2. B
    10
  3. C
    11
  4. D
    9
View written solutionFree

Correct answer: A

  1. Let z=x+iyz=x+iyz=x+iy.

    Given ∣z−2z−3∣=2\left|\frac{z-2}{z-3}\right|=2​z−3z−2​​=2 so ∣z−2∣=2∣z−3∣.|z-2|=2|z-3|.∣z−2∣=2∣z−3∣.

  2. Write in Cartesian form: ∣z−2∣=(x−2)2+y2,∣z−3∣=(x−3)2+y2.|z-2|=\sqrt{(x-2)^2+y^2}, \qquad |z-3|=\sqrt{(x-3)^2+y^2}.∣z−2∣=(x−2)2+y2​,∣z−3∣=(x−3)2+y2​.

    Hence, (x−2)2+y2=2(x−3)2+y2.\sqrt{(x-2)^2+y^2}=2\sqrt{(x-3)^2+y^2}.(x−2)2+y2​=2(x−3)2+y2​.

  3. Squaring both sides: (x−2)2+y2=4((x−3)2+y2).(x-2)^2+y^2=4\big((x-3)^2+y^2\big).(x−2)2+y2=4((x−3)2+y2).

    Expand: x2−4x+4+y2=4x2−24x+36+4y2.x^2-4x+4+y^2=4x^2-24x+36+4y^2.x2−4x+4+y2=4x2−24x+36+4y2.

    Rearranging: 0=3x2−20x+32+3y2.0=3x^2-20x+32+3y^2.0=3x2−20x+32+3y2.

    So, 3x2+3y2−20x+32=0.3x^2+3y^2-20x+32=0.3x2+3y2−20x+32=0.

  4. Divide by 333: x2+y2−203x+323=0.x^2+y^2-\frac{20}{3}x+\frac{32}{3}=0.x2+y2−320​x+332​=0.

    Complete the square in xxx: (x−103)2+y2=1009−323.\left(x-\frac{10}{3}\right)^2+y^2=\frac{100}{9}-\frac{32}{3}.(x−310​)2+y2=9100​−332​.

    (x−103)2+y2=1009−969=49.\left(x-\frac{10}{3}\right)^2+y^2=\frac{100}{9}-\frac{96}{9}=\frac{4}{9}.(x−310​)2+y2=9100​−996​=94​.

  5. Therefore, the circle has center (103,0)\left(\frac{10}{3},0\right)(310​,0) and radius 23.\frac{2}{3}.32​.

    So, α=103,β=0,γ=23.\alpha=\frac{10}{3},\quad \beta=0,\quad \gamma=\frac{2}{3}.α=310​,β=0,γ=32​.

  6. Now compute: 3(α+β+γ)=3(103+0+23)=3⋅123=12.3(\alpha+\beta+\gamma)=3\left(\frac{10}{3}+0+\frac{2}{3}\right)=3\cdot\frac{12}{3}=12.3(α+β+γ)=3(310​+0+32​)=3⋅312​=12.

  7. Hence the correct option is: A: 12\boxed{\text{A: }12}A: 12​

Comparison with stored answer: The stored correct answer is A, which matches the derived answer.

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