JEE Advanced 2021 Paper 1 Q16 Mathematics Algebra Complex Numbers Hard

JEE Advanced 2021 Paper 1 · Q16 · Complex Numbers

For any complex number $w = c + id$, let $\arg(w) \in (-\pi, \pi]$, where $i = \sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z = x + iy$ satisfying $\arg\left(\dfrac{z + \alpha}{z + \beta}\right) = \dfrac{\pi}{4}$, the ordered pair $(x, y)$ lies on the circle $x^2 + y^2 + 5x - 3y + 4 = 0$. Then which of the following statements is (are) TRUE ?

  1. A. $\alpha = -1$
  2. B. $\alpha\beta = 4$
  3. C. $\alpha\beta = -4$
  4. D. $\beta = 4$
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:B, D

Solution

The locus of $z$ with $\arg\left(\dfrac{z+\alpha}{z+\beta}\right)=\dfrac{\pi}{4}$ is an arc of a circle passing through $z=-\alpha$ and $z=-\beta$. The given circle $x^2+y^2+5x-3y+4=0$ has centre $(-5/2,3/2)$ and intersects the real axis where $y=0$: $x^2+5x+4=0$, giving $x=-1,-4$. So the two real points on the circle are $(-1,0)$ and $(-4,0)$, i.e., $-\alpha=-1$ and $-\beta=-4$ (or swap), giving $\alpha=1$, $\beta=4$. Then $\alpha\beta=4$ and $\beta=4$. Hence (B) and (D) are TRUE; (A) is FALSE since $\alpha=1$, not $-1$.

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