JEE Main 2022 · 24 Jun · Shift 2 · Mathematics Q21

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MathematicsInteger+4 / -1Question 21 of 90

Let $S=\{z\in\mathbb C:|z-3|\le1$ and $z(4+3i)+\bar z(4-3i)\le24\}$ . If $\alpha+i\beta$ is the point in $S$ which is closest to $4i$ , then $25(\alpha+\beta)$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 80

The half-plane. $z(4+3i)+\bar z(4-3i)=2\,\mathrm{Re}\,[z(4+3i)]=2(4x-3y)\le24$ , i.e. $4x-3y\le12$ .

Nearest point of the disc. The centre (3, 0) is 5 from (0, 4); the nearest point of $|z-3|\le1$ is $(3,0)+\frac15(-3,4)=(2.4,\ 0.8)$ .

Check. $4(2.4)-3(0.8)=7.2\le12$ , so it lies in $S$ .

Value. $25(\alpha+\beta)=25(3.2)=80$ .

Answer: 80

Topics:Geometry in the complex plane Equation of a circle

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JEE Main 2022 · 24 Jun · Shift 2, Mathematics question 21