## 2018年12月22日土曜日

### 数学 – Python - JavaScript - 数学はここから始まる - 数 – 高次方程式 – 連立2次方程式(消去法、2次方程式の解と係数の関係)

1. $\begin{array}{}y=x-1\\ {x}^{2}+{\left(x-1\right)}^{2}=25\\ 2{x}^{2}-2x-24=0\\ {x}^{2}-x-12=0\\ \left(x-4\right)\left(x+3\right)=0\\ x=-3,y=-4\\ x=4,y=3\end{array}$

2. $\begin{array}{}{x}^{2}+3{x}^{2}=48\\ {x}^{2}=12\\ x=±2\sqrt{3}\\ x=±2\sqrt{3},y=±6\end{array}$

3. $\begin{array}{}{\left(2x-1\right)}^{2}-{x}^{2}=5\\ 4{x}^{2}-4x+1-{x}^{2}-5=0\\ 3{x}^{2}-4x-4=0\\ \left(x-2\right)\left(3x+2\right)=0\\ x=-\frac{2}{3},y=-\frac{4}{3}-1=-\frac{7}{3}\\ x=2,y=4-1=3\end{array}$

4. $\begin{array}{}{t}^{2}-4t+2=0\\ t=2±\sqrt{4-2}\\ =2±\sqrt{2}\\ x=2±\sqrt{2},y=2\mp \sqrt{2}\end{array}$

5. $\begin{array}{}{\left(x+y\right)}^{2}-xy=21\\ 25-xy=21\\ xy=4\\ {t}^{2}-5t+4=0\\ \left(t-1\right)\left(t-4\right)=0\\ x=1,y=4\\ x=4,y=1\end{array}$

6. $\begin{array}{}y=\frac{5-3x}{4}\\ 4{x}^{2}+\frac{5}{4}x-\frac{3}{4}{x}^{2}-3\frac{25+9{x}^{2}-30x}{16}=0\\ 64{x}^{2}+20x-12{x}^{2}-75-27{x}^{2}+90x=0\\ 25{x}^{2}+110x-75=0\\ 5{x}^{2}+22x-15=0\\ \left(x+5\right)\left(5x-3\right)=0\\ x=-5,y=\frac{5+15}{4}=5\\ x=\frac{3}{5},y=\frac{5-\frac{9}{5}}{4}=\frac{25-9}{4·5}=\frac{16}{20}=\frac{4}{5}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, solve, sqrt, plot

print('32.')

x, y = symbols('x, y')

eqss = [(x - y - 1, x ** 2 + y ** 2 - 25),
(y - sqrt(3) * x, x ** 2 + y ** 2 - 48),
(y - (2 * x - 1), y ** 2 - x ** 2 - 5),
(x + y - 4, x * y - 2),
(x + y - 5, x ** 2 + x * y + y ** 2 - 21),
(3 * x + 4 * y - 5, 4 * x ** 2 + x * y - 3 * y ** 2)]

for i, eqs in enumerate(eqss, 1):
print(f'({i})')
pprint(solve(eqs, dict=True))
print()

p = plot(x - 1, sqrt(25 - x ** 2), -sqrt(25 - x ** 2), -4, 3,
legend=True, show=False)
colors = ['red', 'green', 'green', 'blue', 'orange']

for i, color in enumerate(colors):
p[i].line_color = color

p.save('sample32.png')


$./sample32.py 32. (1) [{x: -3, y: -4}, {x: 4, y: 3}] (2) [{x: -2⋅√3, y: -6}, {x: 2⋅√3, y: 6}] (3) [{x: -2/3, y: -7/3}, {x: 2, y: 3}] (4) [{x: -√2 + 2, y: √2 + 2}, {x: √2 + 2, y: -√2 + 2}] (5) [{x: 1, y: 4}, {x: 4, y: 1}] (6) [{x: -5, y: 5}, {x: 3/5, y: 4/5}]$


HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample32.js"></script>


JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
btn0 = document.querySelector('#draw0'),
btn1 = document.querySelector('#clear0'),
input_r = document.querySelector('#r0'),
input_dx = document.querySelector('#dx'),
input_x1 = document.querySelector('#x1'),
input_x2 = document.querySelector('#x2'),
input_y1 = document.querySelector('#y1'),
input_y2 = document.querySelector('#y2'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
p = (x) => pre0.textContent += x + '\n';

let fns = [[x => x - 1, 'red'],
[x => Math.sqrt(25 - x ** 2), 'green'],
[x => -Math.sqrt(25 - x ** 2), 'green']];

let draw = () => {
pre0.textContent = '';

let r = parseFloat(input_r.value),
dx = parseFloat(input_dx.value),
x1 = parseFloat(input_x1.value),
x2 = parseFloat(input_x2.value),
y1 = parseFloat(input_y1.value),
y2 = parseFloat(input_y2.value);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}

let points = [],
lines = [[-3, y1, -3, y2, 'blue'],
[x1, -4, x2, -4, 'blue'],
[4, y1, 4, y2, 'brown'],
[x1, 3, x2, 3, 'brown']];
fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);

points.push([x, y, color]);
}
});

let xscale = d3.scaleLinear()
.domain([x1, x2])
let yscale = d3.scaleLinear()
.domain([y1, y2])

let xaxis = d3.axisBottom().scale(xscale);
let yaxis = d3.axisLeft().scale(yscale);
div0.innerHTML = '';
let svg = d3.select('#graph0')
.append('svg')
.attr('width', width)
.attr('height', height);

svg.selectAll('line')
.data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
.enter()
.append('line')
.attr('x1', (d) => xscale(d[0]))
.attr('y1', (d) => yscale(d[1]))
.attr('x2', (d) => xscale(d[2]))
.attr('y2', (d) => yscale(d[3]))
.attr('stroke', (d) => d[4] || 'black');

svg.selectAll('circle')
.data(points)
.enter()
.append('circle')
.attr('cx', (d) => xscale(d[0]))
.attr('cy', (d) => yscale(d[1]))
.attr('r', r)
.attr('fill', (d) => d[2] || 'green');

svg.append('g')
.attr('transform', translate(0, ${height - padding})) .call(xaxis); svg.append('g') .attr('transform', translate(${padding}, 0))
.call(yaxis);

[fns].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();