【京大2000】二次方程式の複素数解の大きさ【方程式・複素数】

⭐️ "Oshie Math" allows you to ask Hayashi unlimited questions for a fixed monthly fee. You can try asking questions for free once! https://oshiemath.com/ ⭐️ "Hayashi Math Class," a specialized mathematics school where you can receive direct instruction from Hayashi. Please come for a trial lesson and interview! https://hayashi-math.com/ ✅ Official LINE for students applying to top universities: https://lin.ee/lI7n1SJ Subscriber benefits & live streams for applicants ℹ️ Shunsuke Hayashi's profile https://hayashishunsuke.com/profile/ ・Graduated from Sakae Higashi Junior High School → Chikuma High School → University of Tokyo, Faculty of Science, Department of Physics ・Scored 90% on the second-stage mathematics exam at the University of Tokyo, passing the exam as a current student ・2014 Japan Physics Olympiad Gold Medal ・Placed first place in the 2014 University of Tokyo Physics Mock Exam ℹ️ Please note ・The explanations are Shunsuke Hayashi's own and are not official university information. ・Amazon Associates links will be used when introducing books, etc. This problem concerns complex solutions to quadratic equations, taken from the 2000 Kyoto University Science Mathematics exam. In (1), we seek to find the range of the constant a such that a line and (half of) a parabola share a common point. If you can recognize the existence of fixed points through which a line passes regardless of the value of a, this problem will be easy to solve. Not just in this problem, but fixed points through which curves and lines pass regardless of the value of parameters are often useful in solving problems, and are certainly useful at the very least when illustrating graphs. (2) deals with the absolute value of complex solutions to quadratic equations. At first glance, this may seem unrelated to (1), but by dividing both sides by a^2, we can make (1) usable. Since the common point between a line and (half of) a parabola corresponds to the real solution, the larger x-coordinate of the common point is β. However, (1) can only be used when a real solution exists. Note that a separate discussion is required for imaginary solutions. Use (1) to find the minimum value of |β| when there is a real solution, and prove that |β| is greater when there is an imaginary solution. Since (2) can also have an imaginary solution, many people get confused and stop working on it. However, since no special techniques are required, this is a challenging problem that makes a difference. --------- Table of Contents 00:00 2000 Kyoto University Science Mathematics [2] 00:45 (1) C: Graph of y = √x 02:12 (1) Are there any fixed points through which the line l passes? 03:48 (1) What is the range of the slope of the line l? 06:04 (1) Image of the answer 07:44 (1) There is no intersection when x ≥ 1/2 10:11 (1) When it passes through the origin and when it is tangent 14:54 (1) The range of a that satisfies the condition 16:47 (1) Summary of the solution 19:05 (2) What is the relationship with (1)? 20:19 (2) When equation (★) has a real solution 26:45 (2) Summary of the case when it has a real solution 28:23 (2) When equation (★) has an imaginary solution 32:05 (2) Summary of the conclusion and solution 35:01 (2) Conclusion