JEE Advanced 2021 · Paper 1 · Mathematics Q16

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MathematicsMultiple correct+4 / -2Question 54 of 57

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 $\arg\left(\dfrac{z + \alpha}{z + \beta}\right) = \dfrac\pi4$ , 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?

\alpha = -1

\alpha\beta = 4

\alpha\beta = -4

\beta = 4

Answer and worked solution

Official answer: B, D

The locus. $\arg\dfrac{z+\alpha}{z+\beta}=\dfrac\pi4$ is an arc of a circle through $-\alpha$ and $-\beta$ .

Those points. The given circle meets the real axis where $x^2+5x+4=0$ , i.e. at $-1$ and $-4$ .

Which is which. Test the top point of the circle, $z=-2.5+3.62i$ (centre $(-2.5,1.5)$ , radius $\approx2.12$ ):

$\arg(z+1)\approx112.5^\circ$ ;
$\arg(z+4)\approx67.5^\circ$ .

The difference is $+45^\circ$ , so $\alpha=1$ and $\beta=4$ . Then $\alpha\beta=4$ and $\beta=4$ . (B), (D)

Topics:Geometry in the complex plane Equation of a circle

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JEE Advanced 2021 · Paper 1, Mathematics question 16