my scientific age

Chapter 16 Solving the Area of ​​a Hyperbolic Focal Triangle
Back in Ping'an, I found a restaurant to invite He Lao and his party to dinner.

Three rounds of wine, five flavors of food.

After eating, it was late at night.

A group of people left the restaurant and arrived at the intersection of Ping'anli. Yu Hua and Mrs. He said goodbye to Mr. He and asked about the fee. The tomb was 35 yuan, and the funeral incense and candle paper money was 45 yuan. The rest of the fee was not charged, a total of [-] yuan.

Agreed to come to the door to settle the expenses tomorrow, say goodbye to He Lao and others, and the two went to Jinguo Hutong.

When I got home, entered the door, the main hall returned to its former appearance, and the portrait of Yu Qinghe was placed in the middle.

He took out three sticks of incense and lit them, put them on the incense table, bowed three times, and 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 is over, and Yu Hua can finally spare a lot of time to concentrate on his studies.

Learning always comes first.

Time doesn't wait for me, just fight for the day and night.

He has no time to waste.

"Master, won't you take a break?" Xu Rui looked at Yu Hua with a tired face and asked with concern.

"Do not bother me."

Yu Hua waved his hand, dragged his tired body, and walked to the bedroom step by step, entered the door and pulled the switch on the straw rope, turned on the light, sat at the desk and opened the textbook on arithmetic, and returned to the place where he stayed in the morning - Analytical Geometry.

Analytical Geometry.

This is a combination of intelligence and difficulty, and students generally call it the most obscene stable problem.

There is no way, the problem of analytic geometry, whether it is the simplest straight line, or the moderately difficult triangle and circle, the calculation process is extremely complicated, and the amount of calculation is huge, and it is deduced layer by layer. If any calculation steps are wrong, you cannot continue to write.

Waste of spirit, waste of ink, and waste of draft paper.

This is a well-known key problem in high school. When he took the college entrance examination in later generations, Yu Hua had a headache and gave up immediately.

With physical fatigue and a heart of seeking knowledge, Yu Hua sank into learning.

Analytical geometric straight line, the first rule——

The angle of inclination and the slope of the straight line.

Inclination angle: the angle between the upward direction of the straight line and the positive direction of the X axis, usually denoted as α, and the range is [0, π); when the straight line is a horizontal line, α=0 is specified.

The slope is also known as the angle coefficient, which is the amount of the inclination of a straight line to the abscissa axis in the plane rectangular coordinate system. When the tangent of the inclination angle is K=tanα; when α=π/2, the slope of the straight line does not exist;
When the straight line l is parallel or coincident with the X axis, α=0 is specified, when α≠π/2, the slope K=tanα, and when α=π/2, the slope K does not exist.

The key point to note is that each straight line has a correct inclination angle, which reflects the inclination of the straight line to the positive X axis...

Carefully read the knowledge points about the basic stage of analytic geometry in the early stage, although his body was exhausted, Yu Hua still quickly entered a familiar state of ecstasy.

The whole person is extremely focused, as if not disturbed by any foreign objects. The complex and obscure knowledge points are gradually understood, transformed into three-dimensional and intuitive mathematical symbols in the mind, and then evolved into mathematical formulas according to the laws.

This is an incomprehensible pleasure for ordinary people. Yu Hua only feels that he is swimming in the sea of ​​mathematics, swimming happily like a dolphin, sometimes turning in circles, sometimes floating on the surface and spitting out a mouthful of water vapor, and then suddenly jumping to the bottom of the sea.

Comfortable.

carefree.

There was even a hint of pleasure.

The straight line content of analytic geometry is easy and relaxed, the circle of analytic geometry is striding forward, the ellipse of analytic geometry is small, and the hyperbola of analytic geometry...

Combined with the math knowledge he had learned from his predecessor, now, Yu Hua's learning efficiency and progress are extremely objective.

I don't know how long has passed.

The cold wind whistled outside the window, and it was extremely cold inside.

His eyes were fixed on the hyperbolic problem in front of him. Yu Hua's face was serious, his brows were slightly wrinkled, a layer of sweat was oozing from his forehead, and he was no longer in high spirits. This was a very difficult hyperbolic problem.

已知双曲线x/9-y/16=1的左、右焦点分为别F1F2，若双曲线上一点P使∠F1PF2=90°，则△F1PF的面积是多少。

The main content is to solve the area of ​​hyperbolic focal triangle. It is a textbook topic compiled by Princeton University professors for middle school students. The area formulas and principles are not difficult.

Yu Hua had already counted it four times, and he had already piled up more than ten pages of scratch paper on the desk, but he still couldn't count it.

It's not that the calculated answer is wrong, it's that it doesn't count at all.

"Strange, could it be that there is something wrong with my thinking? To solve it from another angle, it seems like this..." Yu Hua rubbed his slightly swollen forehead, held a pencil in his right hand, and counted again.

According to the hyperbolic focal triangle formula S=bcot(θ/2), according to the definition of hyperbola: ‖PF1|-|PF2‖=6.

两边平方得：｜PF1｜+｜PF2｜-2｜PF1‖PF2｜=36。

According to the Pythagorean theorem:
∵，｜PF1｜+｜PF2｜=｜F1F2｜=100
∴, |PF1‖PF2|=32
∴，S=1/2（｜PF1‖PF2｜）=16。

"Huh, it seems right, it should be sixteen, and I finally figured it out." Yu Hua put down his pencil and looked at the densely packed scratch paper. He finally breathed a sigh of relief. He reached out and wiped the sweat from his forehead, feeling a sense of accomplishment in his heart. However live.

It became.

I have basically mastered the hyperbolic focal triangle that I hated and disliked the most in the past. Today, the progress of the math textbook has been greatly improved, which is gratifying.

After resting for half a minute, Yu Hua did not continue to write and study. He had already withdrawn from the state of extremely focused self-absorption, and took another look at the mathematics textbook. Sure enough, the above series of knowledge points had all become obscure and abstract, and it was difficult to understand for a while.

Taking another look at the scratch paper, the hyperbolic focal triangle problem written on it became obscure and difficult to understand. The entire calculation formula and process made Yu Hua dazzled, and it was far from the state of divine assistance half a minute ago.

Hey, what's the area?
Wait, how do the left and right focus F1 and F2 count?

Looking at it twice, Yu Hua felt that his head was a little confused. He threw away the pencil, chose a game, and raised his hand to look at his watch. It was 11:30 in the middle of the night, four hours had passed, and he thought in his heart: "I have reached the limit, my head is unresponsive. A feeling of lack of oxygen, four hours of study time, plus 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 the present, Yu Hua discovered that the self-absorbed state of learning occurs when the brain is normal. In this state, he feels as if he has mastered everything, and is in the world of knowledge, enjoying the knowledge that comes from knowledge. Baptism and indoctrination, all kinds of inspiration continue to emerge, which can make him feel the joy of mathematics.

But as the brain gradually overuses and becomes tired, until the hypoxia reaches its limit, it will withdraw from this state.

At this time, the joy of mathematics will be reversed to the torture of mathematics in an instant.

What pleasure and comfort?

Go aside.

After tonight's test, this state of ecstasy can last about six hours a day.

(End of this chapter)