Let <img alt="$k$" height="12" src="https://latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png" width="9" /> be a real number such that the system                                                                 |25+20i−z|=5                                                              |z−4−k|=|z−3i−k| has exactly one complex solution <img alt="$z$" height="10" src="https://latex.artofproblemsolving.com/b/1/3/b13f21416d84e13708696f34dea81026cda583c9.png" width="11" />. The sum of all possible values of <img alt="$k$" height="12" src="https://latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png" width="9" /> can be written as <img alt="$\frac{m}{n}$" height="33" src="https://latex.artofproblemsolving.com/2/7/f/27f583697295fc6f723862c13228ee55107c33ca.png" width="18" />, where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. Find <img alt="$m + n$" height="12" src="https://latex.artofproblemsolving.com/5/3/b/53bf1afcb274a3bb84338109fffec8a1f1c02008.png" width="48" />. Here <img alt="$i = \sqrt{-1}$" height="18" src="https://latex.artofproblemsolving.com/d/a/c/dac5dec897250384833e3c1b44639bf91cca823a.png" width="69" />.

Question No: 329

Let $k$ be a real number such that the system

|25+20i−z|=5

|z−4−k|=|z−3i−k|

has exactly one complex solution $z$ . The sum of all possible values of $k$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . Here $i = \sqrt{-1}$ .

Editorial

Tags: Algebra Geometry

Question Details