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).

Answer 1

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.