The answers to the questions about this orbit system are as follows:
- This is a standard ellipse
- The given points in the question are 2.667 and -3.99
How to solve the system of equation
We have the following equation
[tex]16x^{2} + 192x + 576 + 25y^{2} = 1600[/tex]
we can factorize as:
= [tex]16(x^{2} + 12x + 36) + 25y^{2} = 1600[/tex]
[tex]= 16(x + 6) (x + 6) + 25y^{2} = 1600[/tex]
[tex]= 16(x+6) ^{2} + 25(y-0)^{2} = 1600[/tex]
[tex]= \frac{(x+6) ^{2}}{10^{2} } + \frac{(y-0)^{2}}{8^{2} } = 1[/tex]
The equation that we have above is the standard ellipse equation.
2.
[tex]16x^{2} -96x+144+ 25y^{2} = 400[/tex]
[tex]= 16(x^{2} -6x+9)+ 25y^{2} = 400[/tex]
[tex]= 16(x^{2} -3x-3x+9)+ 25y^{2} = 400[/tex]
[tex]= 16(x -3)(x-3))+ 25y^{2} = 400[/tex]
[tex]= 4^{2} (x-3)^{2} + 5^{2} (y-0)^{2} = 400[/tex]
[tex]= \frac{(x-3)^{2}}{5^{2}} + \frac{4^{2} }{4^{2}} = 1[/tex]
The equation that we have above is the standard ellipse equation.
We can see that the semi major axis of the first equation gives us 10 units while that of the second gives us 5 units. The first equation has the larger orbit than the second.
3. (x + 6)²/100 + y²/64
= x-3²/25 + y²/16
[tex]16x^{2} +192x+576+25y^{2} = 1600[/tex]
[tex]16x^{2} -96x+144+ 25y^{2} = 400[/tex]
solve both equations using the simultaneous linear equation
the values for x and y are
2.667 and -3.99
Complete question
4. Gravity holds objects in space together. It makes the moon orbit our planet, and it makes the planets in our solar system orbit the sun. In some parts of space, gravity can make two s All changes saved orbit each other, and when this happens, we call the pair "binary stars." In reality, both stars orbit a center of mass outside of their individual masses, as shown below:
orbit of less
orbit of more
For the above system, we place the center of mass at the origin. The equations for the two stars' orbits are:
16x2+192x+576+25y² = 1600
16296x+144+ 25y² = 400
a) Determine whether the equations represent parabolas, hyperbolas, or ellipses.
b) Write both equations in standard form, and show how you arrived at your answers. Which equation matches with each orbit? How do you know? c) Although these two stars are unlikely to crash, at how many points on their orbits would such an event be possible? Explain how you would find those
Read more on standard ellipse equation here: https://brainly.com/question/10259559
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