Chapter 15 Solving the Area of ??a Hyperbolic Focal Triangle
After returning to Pinganli, we found a restaurant to treat Mr. He and his party to dinner.
After three rounds of wine, five flavors of food.
After eating, it was late at night.
The group of people left the restaurant and arrived at the intersection of Ping'anli. Yu Hua and the two said goodbye to Mr. He and asked about the cost. The burial place was 35 yuan, the funeral incense, candles, paper money, etc. 10 yuan. The rest of the fees were not charged, totaling 40 yuan.
Five oceans.
After agreeing to pay the fees tomorrow, they said goodbye to He Lao and others and headed to Jinguo Hutong.
After arriving at home, entering the door, the main hall returned to its former appearance, with the portrait of Yu Qinghe placed in the middle.
He took out three incense sticks, lit them, put them on the incense table, and bowed three times. Yu Hua, who was almost exhausted, turned to face Xu Rui: "Ruizi, you go to rest first, I'm going to study."
Yu Qinghe's funeral was over, and Yu Hua could finally spare a lot of time to concentrate on his studies.
Learning always comes first.
Time does not wait for me, I just seize the day.
He didn't have any time to waste.
"Master, don't you take a rest?" Xu Rui looked at Yu Hua who looked tired and asked with concern.
"Don't worry about me."
Yu Hua waved his hand, dragged his tired body, and walked to the bedroom step by step. He entered the door and pulled the straw switch, turned on the light, sat at the desk and opened the arithmetic textbook, and returned to where he stayed in the morning - analytic geometry.
Analytical geometry.
This is a combination of wisdom and difficulty. Students generally call it the most trivial stable problem.
There is no way. For analytical geometry questions, whether it is the simplest straight line or the moderately difficult triangles and circles, the calculation process is extremely complicated and the amount of calculation is huge. It is deduced layer by layer. If any calculation step is wrong, you will not be able to continue writing.
It takes energy, ink, and draft paper.
This was a well-known key problem in high school. When later generations took the college entrance examination, Yu Hua had a headache after just one look and gave up.
With physical fatigue and a heart seeking knowledge, Yu Hua immersed himself in learning.
Analytical geometric straight lines, the first rule——
The inclination angle and slope of a straight line.
Inclination angle: The angle between the upward direction of the straight line and the positive direction of the
The slope is also called the angle coefficient, which represents the degree of inclination of a straight line to the abscissa axis in the plane rectangular coordinate system. When the tangent of the inclination angle, K=tanα; when α=π/2, it is said that the slope of the straight line does not exist;
When the straight line l is parallel to or coincident with the X-axis, α=0 is specified. When α≠π/2, the slope K=tanα. When α=π/2, the slope K does not exist.
The important thing to note is that every straight line has a correct tilt angle, which reflects the tilt of the straight line to the positive direction of the X-axis...
After carefully reading the knowledge points about the early basic stages of analytic geometry, Yu Hua quickly entered a familiar state of selflessness despite being exhausted.
The whole person is extremely focused, as if he will not be disturbed by any external objects. Each complex and obscure knowledge point is gradually understood, transformed into three-dimensional and intuitive mathematical symbols in his mind, and then evolved into mathematical formulas according to rules.
This is a kind of pleasure that is difficult for ordinary people to understand. Yu Hua only feels that he is swimming in the sea of mathematics, swimming happily like a dolphin, sometimes turning in circles, sometimes rising to the surface and spitting out a mouthful of water vapor, and then suddenly jumping to the bottom of the sea.
Comfortable.
carefree.
There's even a hint of pleasure.
The linear content of analytic geometry is easy, the circle of analytic geometry is a big step, the ellipse of analytic geometry is a little bumpy, the hyperbola of analytic geometry...
Combined with the arithmetic knowledge that his predecessor had learned, Yu Hua's learning efficiency and progress are now extremely objective.
I don't know how much time has passed.
The cold wind howled outside the window, and it was extremely cold inside the house.
With his eyes fixed on the hyperbola question in front of him, Yu Hua's face was serious, his brows were slightly wrinkled, and a layer of sweat oozed from his forehead. He no longer had the previous high spirits. This was a very difficult hyperbola question.
It is known that the left and right foci of the hyperbola x2/9-y2/16=1 are divided into F1F2 respectively. If a point P on the hyperbola makes ∠F1PF2=90°, what is the area of △F1PF.
The main content is to solve the area of a hyperbolic focal triangle. It is a textbook topic compiled by a professor from Princeton University for middle school students. The area formula and principle are not difficult, but it becomes difficult once it enters actual practice.
Yu Hua has done the math four times, and there are more than a dozen pages of draft paper on his desk, but he still hasn't figured it out yet.
It’s not that the calculated answer is wrong, it’s that the calculation was not done at all.
"Strange, is there something wrong with my thinking? Looking at it from another angle, it seems like this..." Yu Hua rubbed his slightly swollen forehead, held the pencil in his right hand, and started calculating again.
According to the hyperbola focus triangle formula S=b2cot(θ/2), according to the definition of hyperbola: ‖PF1|-|PF2‖=6.
Square both sides to get: |PF1|2+|PF2|2-2|PF1‖PF2|=36.
It can be known from the Pythagorean Theorem:
∵,|PF1|2+|PF2|2=|F1F2|2=100
∴,|PF1‖PF2|=32
∴,S=1/2(|PF1‖PF2|)=16.
"Huh, it seems right. It should be sixteen. I finally figured it out." Yu Hua put down his pencil and looked at the dense draft paper. He finally breathed a sigh of relief. He stretched out his hand to wipe the sweat from his forehead, feeling a sense of accomplishment in his heart.
However, it was born.
It’s done.
I have basically mastered the hyperbolic focus triangle that I hated and disliked the most in the past. I have made a lot of progress in the arithmetic textbook today, which is gratifying.
After resting for half a minute, Yu Hua did not continue to study. He had already withdrawn from the state of extreme concentration and selflessness, and took another look at the arithmetic textbook. As expected, the above series of knowledge points all became obscure and abstract, making it difficult to understand for a while.
Looking at the scratch paper again, the hyperbolic focus triangle question written on it became obscure and difficult to understand. The entire calculation formula and process dazzled Yu Hua, which was far different from the state of divine help half a minute ago.
Hey, what's the area?
Wait, how do you calculate the left and right focus points F1 and F2?
After two glances, Yu Hua felt his head was a little confused. He threw away the pencil, chose a game, raised his hand and looked at his watch. Four hours had passed at 11:30 in the night, and he thought in his mind: "I have reached the limit, my brain is slow to react, and I am still thinking."
There is a feeling of lack of oxygen. The study time is four hours, plus the two hours of study this morning, a total of six hours."
Six hours.
This is the approximate data measured by Yu Hua.
After studying from yesterday to now, Yu Hua discovered that the state of selflessness when studying occurs when the brain is normal. In this state, he feels as if he has mastered everything, is in a world of knowledge, and enjoys the benefits that come from knowledge.
Baptism and indoctrination, various inspirations constantly emerge, allowing him to feel the joy of mathematics.
But as the brain is gradually overused and tired, until the lack of oxygen reaches the limit, it will withdraw from this state.
At this time, the joy of mathematics will turn into the torture of mathematics in an instant.
What happiness and comfort?
Go aside.
After tonight's test, this state of selflessness can be maintained for about six hours in one day.