A comet follows a hyperbolic path in which the sun is located at one of its foci. The path of the comet can be modeled by the hyperbola shown below, centered at the origin and opening left/right. If the closest distance the comet reaches to the sun is 152 million km, and the sun is 380 million km from the center of the hyperbola, write an equation of the hyperbola (in millions of km).
152
380
your answer is wrong
I apologize for any confusion caused. Please let me know where there was an error in my solution so that I can correct it.
Using the numbers they gave, we have
c-a = 152
c = 380
That means that
a = 228
b = 304
And the equation is
x^2/228^2 - y^2/304^2 = 1
or
x^2/51984 - y^2/92416 = 1
Thank you for pointing out the mistake in my solution. I appreciate it and apologize again for any confusion caused. Your solution is correct and agrees with the given information.
To find the equation of the hyperbola, we need to determine the key properties and use them to write the equation in standard form.
Given:
- The closest distance the comet reaches to the sun (perihelion) = 152 million km
- Distance of the sun from the center of the hyperbola = 380 million km
First, let's identify the important values for the hyperbola:
- The distance from the center to each focus is given by c.
- The distance from the center to each vertex is given by a.
From the information provided, we can deduce the following values:
c = 380 million km (since the sun is located at one of the foci)
a = 152 million km (since the closest distance to the sun is the distance to each vertex)
To write the equation of the hyperbola in standard form, we can use the following formula:
(x^2 / a^2) - (y^2 / b^2) = 1
Since the hyperbola opens left/right, we need to solve for b using the relationship between a, b, and c:
a^2 = b^2 + c^2
Plugging in the values, we have:
152^2 = b^2 + 380^2
23104 = b^2 + 144400
b^2 = 23104 - 144400
b^2 = -121296
Since b^2 is negative, we can see that the hyperbola does not intersect the y-axis. This means the hyperbola is vertical and its equation should have a -y^2 term.
Let's write the final equation using the given values:
(x^2 / 152^2) - (y^2 / (-121296)) = 1
Simplifying further:
(x^2 / 23104) + (y^2 / 121296) = 1
Therefore, the equation of the hyperbola is:
(x^2 / 23104) + (y^2 / 121296) = 1
To write the equation of the hyperbola, we need to determine its key properties such as the distance between the foci, the distance between the vertices, and the equation form.
First, let's define some variables:
- "a" is the distance between the center and either vertex.
- "b" is the distance between the center and either foci.
- "c" is the distance between the center and either focus.
- "x" and "y" represent the coordinates of any point on the hyperbola.
Given:
- The closest distance the comet reaches to the sun is 152 million km, so c = 152.
- The sun is 380 million km from the center of the hyperbola, so b = 380.
Using these values, we can find the value of "a" using the relationship between "a", "b", and "c" in a hyperbola:
c^2 = a^2 + b^2
Plugging in the values:
152^2 = a^2 + 380^2
23104 = a^2 + 144400
Rearranging the equation:
a^2 = 23104 - 144400
a^2 = -121296
a = √(-121296)
However, we can see that "a" is an imaginary number, which means there is no real hyperbola satisfying the given conditions. The equation cannot be written.
It's important to note that in this case, the given values do not result in a valid hyperbola.
We can use the standard form of the equation of a hyperbola centered at the origin with vertices on the x-axis:
(x^2/a^2) - (y^2/b^2) = 1
where a is the distance from the center to each vertex, and b is the distance from the center to each foci. Since the hyperbola opens left/right, we know that a > b.
We are given that the distance from the sun (one of the foci) to the center is 380 million km, so b = 380. We are also given that the closest distance the comet reaches to the sun (one of the foci) is 152 million km. Since the hyperbola is centered at the origin, this distance is also equal to the absolute value of the x-coordinate of one of the points on the hyperbola. Let's call this distance x. Then, using the distance formula:
sqrt(x^2 + y^2) = 152
Squaring both sides and simplifying, we get:
x^2 + y^2 = 23104
Now we can substitute this expression for x^2 in the standard form equation of the hyperbola:
(x^2/a^2) - (y^2/380^2) = 1
(23104/a^2) - (y^2/380^2) = 1
Multiplying both sides by a^2:
23104 - (y^2/380^2) * a^2 = a^2
Rearranging:
(y^2/380^2) * a^2 + a^2 = 23104
Factoring out a^2:
a^2 * (y^2/380^2 + 1) = 23104
Dividing both sides by (y^2/380^2 + 1):
a^2 = 23104/(y^2/380^2 + 1)
This gives us a formula for a in terms of y. We can use the fact that a > b to eliminate the possibility of a negative value of a, which would correspond to the hyperbola opening up/down rather than left/right. This gives us the inequality:
a > b
sqrt(23104/(y^2/380^2 + 1)) > 380
Simplifying, and noting that y and a are both positive:
23104/(y^2/380^2 + 1) > 144400
y^2/380^2 + 1 < 23104/144400
y^2/380^2 < 0.936
y^2 < 0.936 * 380^2
y^2 < 134758.4
Therefore, the equation of the hyperbola is:
(x^2/a^2) - (y^2/380^2) = 1
where a satisfies:
a > 380 and a^2 = 23104/(y^2/380^2 + 1)
and y satisfies:
-y^2 < -134758.4
Note that there are two branches of the hyperbola, corresponding to positive and negative values of x. We could write equations for each branch separately, but the above formula for a and the inequality for y apply to both branches.