Question 1209799: 1. 137+6y=-y^2-x^2-24x
2. x^2+y^2+14x-12y+4=0
3. y^2+2x+x^2=24y-120
4. x^2+2x+y^2=55+10y
5. 8x+32y+y^2=-263-x^2
Found 2 solutions by timofer, CPhill:
Answer by timofer(104)   (Show Source):
You can put this solution on YOUR website! Use the previous response which was for your equation 1 here. You want to complete the squares if necessary and put equation into the needed form. Decide then what it is or make the graph.
Answer by CPhill(1959)  (Show Source):
You can put this solution on YOUR website! The provided equations are not in the standard form of a circle, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. I'll first convert them to the standard form and then plot them using Altair.
1. **137 + 6y = -y^2 - x^2 - 24x**
Rearranging the terms, we get x^2 + 24x + y^2 + 6y = -137
Completing the square for x, we add (24/2)^2 = 144 to both sides: x^2 + 24x + 144 + y^2 + 6y = -137 + 144
Completing the square for y, we add (6/2)^2 = 9 to both sides: x^2 + 24x + 144 + y^2 + 6y + 9 = -137 + 144 + 9
Simplifying, we get (x + 12)^2 + (y + 3)^2 = 16 = 4^2
So, the center is (-12, -3) and the radius is 4.
2. **x^2 + y^2 + 14x - 12y + 4 = 0**
Rearranging the terms, we get x^2 + 14x + y^2 - 12y = -4
Completing the square for x, we add (14/2)^2 = 49 to both sides: x^2 + 14x + 49 + y^2 - 12y = -4 + 49
Completing the square for y, we add (-12/2)^2 = 36 to both sides: x^2 + 14x + 49 + y^2 - 12y + 36 = -4 + 49 + 36
Simplifying, we get (x + 7)^2 + (y - 6)^2 = 81 = 9^2
So, the center is (-7, 6) and the radius is 9.
3. **y^2 + 2x + x^2 = 24y - 120**
Rearranging the terms, we get x^2 + 2x + y^2 - 24y = -120
Completing the square for x, we add (2/2)^2 = 1 to both sides: x^2 + 2x + 1 + y^2 - 24y = -120 + 1
Completing the square for y, we add (-24/2)^2 = 144 to both sides: x^2 + 2x + 1 + y^2 - 24y + 144 = -120 + 1 + 144
Simplifying, we get (x + 1)^2 + (y - 12)^2 = 25 = 5^2
So, the center is (-1, 12) and the radius is 5.
4. **x^2 + 2x + y^2 = 55 + 10y**
Rearranging the terms, we get x^2 + 2x + y^2 - 10y = 55
Completing the square for x, we add (2/2)^2 = 1 to both sides: x^2 + 2x + 1 + y^2 - 10y = 55 + 1
Completing the square for y, we add (-10/2)^2 = 25 to both sides: x^2 + 2x + 1 + y^2 - 10y + 25 = 55 + 1 + 25
Simplifying, we get (x + 1)^2 + (y - 5)^2 = 81 = 9^2
So, the center is (-1, 5) and the radius is 9.
5. **8x + 32y + y^2 = -263 - x^2**
Rearranging the terms, we get x^2 + 8x + y^2 + 32y = -263
Completing the square for x, we add (8/2)^2 = 16 to both sides: x^2 + 8x + 16 + y^2 + 32y = -263 + 16
Completing the square for y, we add (32/2)^2 = 256 to both sides: x^2 + 8x + 16 + y^2 + 32y + 256 = -263 + 16 + 256
Simplifying, we get (x + 4)^2 + (y + 16)^2 = 9 = 3^2
So, the center is (-4, -16) and the radius is 3.
Now, I'll use Altair to plot these circles.
```python
import altair as alt
import pandas as pd
import numpy as np
# Define the center and radius of each circle
circles = [
{'center': (-12, -3), 'radius': 4, 'name': 'Circle 1'},
{'center': (-7, 6), 'radius': 9, 'name': 'Circle 2'},
{'center': (-1, 12), 'radius': 5, 'name': 'Circle 3'},
{'center': (-1, 5), 'radius': 9, 'name': 'Circle 4'},
{'center': (-4, -16), 'radius': 3, 'name': 'Circle 5'}
]
# Create a list to hold the data for all circles
all_circles_data = []
for circle in circles:
# Generate points on the circle circumference
theta = np.linspace(0, 2 * np.pi, 100)
x_circ = circle['center'][0] + circle['radius'] * np.cos(theta)
y_circ = circle['center'][1] + circle['radius'] * np.sin(theta)
# Create a DataFrame for the circle data
circle_data = pd.DataFrame({
'x': x_circ,
'y': y_circ,
'Circle': circle['name']
})
all_circles_data.append(circle_data)
# Concatenate all circle DataFrames into a single DataFrame
all_circles_df = pd.concat(all_circles_data)
# Create the Altair chart
chart = alt.Chart(all_circles_df).mark_circle().encode(
x=alt.X('x', axis=alt.Axis(title='X-axis')),
y=alt.Y('y', axis=alt.Axis(title='Y-axis')),
color='Circle', # Use color to differentiate circles
tooltip=['Circle', 'x', 'y']
).properties(
title='Circle Plots'
)
# Save the chart
chart.save('circle_plots.json')
```
This code will generate a JSON file that can be opened in any web browser to view the interactive plot of the circles.
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