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).
Since the hyperbola opens left/right, we know that the x-term should be positive. We also know that the distance between the center and one of the foci (in this case, the sun) is 380 million km, and the distance between the center and a point on the hyperbola (the closest distance the comet reaches to the sun) is 152 million km.
Using the formula for a hyperbola, we can write:
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
where $a$ represents the distance between the center and either vertex, and $b$ represents the distance between the center and a point on the hyperbola.
To find $a$, we use the fact that the distance between the two vertices of a hyperbola is $2a$. In this case, the distance between the two vertices is equal to the distance between the two points on the hyperbola where it intersect the x-axis. We can find this distance by subtracting the x-coordinates of these two points:
$2a = (-152) - (152) = -304$
Dividing by 2 gives us $a = -152$.
To find $b$, we use the fact that $c$, the distance between the center and one focus, is equal to $\sqrt{a^2 + b^2}$. Solving for $b$, we get:
$b = \sqrt{c^2 - a^2} = \sqrt{(380)^2 - (-152)^2} = \sqrt{171360} \approx 414.2$
Thus, we can write the equation of the hyperbola in millions of km as:
$\frac{x^2}{152^2} - \frac{y^2}{414.2^2} = 1$
Note that we could have also written the equation with $a$ and $b$ as coefficients:
$\frac{(x-0)^2}{(-152)^2} - \frac{(y-0)^2}{414.2^2} = 1$
but since the center is at the origin, the $-0$ doesn't contribute anything and the equation simplifies to the one shown above.
To write the equation of a hyperbola, we need the coordinates of the center, the distance between the center and the focus, and the distance between the center and a vertex.
Given:
- The sun is located at one of the foci of the hyperbola.
- The closest distance the comet reaches to the sun is 152 million km.
- The sun is 380 million km from the center of the hyperbola.
First, let's find the distance between the center and a vertex.
The distance between the center and the focus is given by the equation |c| = ae, where a is the distance between the center and a vertex, c is the distance between the center and the focus, and e is the eccentricity of the hyperbola.
In this case, c = 380 million km, and we need to find a.
We know that the distance between the focus and a vertex, a, can be found using the equation a = √(c^2 - b^2), where b is the closest distance the comet reaches to the sun.
Substituting the given values into the equation, we have:
a = √((380 million km)^2 - (152 million km)^2)
a = √(144400 million km^2 - 23104 million km^2)
a = √(121296 million km^2)
a ≈ 348.42 million km
The distance between the center and a vertex is approximately 348.42 million km.
Since the hyperbola opens left/right, the equation will have the form:
(x^2 / a^2) - (y^2 / b^2) = 1
Substituting the values we found, the equation of the hyperbola is:
(x^2 / (348.42 million km)^2) - (y^2 / (152 million km)^2) = 1
Therefore, the equation of the hyperbola (in millions of km) centered at the origin and opening left/right is:
(x^2 / 121296 million km^2) - (y^2 / 23104 million km^2) = 1