Question

Your mission is to track incoming meteors to predict whether or not they will strike Earth. Since Earth has a circular cross-section, you decide to set up a coordinate system with its origin at Earth's center. The equation of Earth's surface is x^2+y^2=40.68, where x and y are distances in thousands of kilometers.

You observe a meteor moving along a path from left to right whose equation is
(240/361)(y-15)^2-x^2=60
, where y <= 5.5. What conic section does the path of the meteor travel?

The meteor's path is a(n) .

What is the center of the conic section that describes the path of the meteor? (,)

Determine if the meteor will strike the earth and if it does strike the earth at what point.

(,)

Answer 1

The meteor's path is a hyperbola.

The center of the conic section is at the point (0, 15).

The meteor will not strike Earth.

1. Identify the Conic Section:

To identify the conic section, we need to rewrite the given equation:

(240/361)(y-15)^2 - x^2 = 60

Moving the constant term:

(240/361)(y-15)^2 = x^2 + 60

Dividing both sides by 60 and rearranging:

(y-15)^2 / (361/240) - x^2 / 60 = 1

This resembles the standard equation of a hyperbola:

(y-k)^2 / a^2 - (x-h)^2 / b^2 = 1

Therefore, the meteor's path is a hyperbola.

2. Locate the Center:

From the standard equation, we can extract the coordinates of the center (h, k):

Center (h, k) = (0, 15)

3. To determine if the meteor will strike Earth, we need to check if the hyperbola intersects the circle representing Earth's surface.

Earth's center: (0, 0)

Earth's radius (d): √40.68 ~ 6.4 km

We know the hyperbola passes through the point (0, 15). To check for intersection, we can compare the minimum distance of the hyperbola from its center (focal length) to the Earth's radius:

Focal length of a hyperbola: b√(a^2 + b^2)

Calculating the focal length for the given hyperbola:

a = √(361/240) ~ 1.2

b = √60 ~ 7.7

Focal length: 7.7 * √(1.2^2 + 7.7^2) ~ 62.2 km

Since the calculated focal length (62.2 km) is greater than Earth's radius (6.4 km), the hyperbola won't intersect the Earth's surface. Therefore, the meteor will not strike Earth.