<p>A region <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" /> in the complex plane is defined by<img alt="\[S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.\]" height="18" src="https://latex.artofproblemsolving.com/1/f/1/1f183c6f855fb40cc0df24c6d24b6a99b1218b0d.png" width="310" />A complex number <img alt="$z = x + iy$" height="16" src="https://latex.artofproblemsolving.com/5/6/0/5600c4ebdb9f6c9949c1486075fd832e16b295f9.png" width="81" /> is chosen uniformly at random from <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" />. What is the probability that <img alt="$\left(\frac34 + \frac34i\right)z$" height="43" src="https://latex.artofproblemsolving.com/b/c/f/bcfe040e6a7a03be317a8fd349c739cb4ef11687.png" width="92" /> is also in <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" />?</p>

Question No: 290

A region $S$ in the complex plane is defined by $S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.$ A complex number $z = x + iy$ is chosen uniformly at random from $S$ . What is the probability that $\left(\frac34 + \frac34i\right)z$ is also in $S$ ?

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Tags: Algebra Geometry Probability

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