Question

the path of a ship can be described by a hyperbolic model centered at the origin relative to two stations on the shore 168 miles apart they are located at the foci. if the ship is 40 miles south of the centre of the hyperbola find the equation of the hyperbola

Answer 1

Answer with explanation:

The focus of hyperbola, which is represented by two stations which are equidistant from center is 168 miles.

So, the coordinates of focus will be C(84,0) and C'(-84,0).

Hyperbola is defined as locus of all the points in the plane, such that it's distance from two fixed points called focus is always constant equal to a constant greater than zero.

The general equation of hyperbola is,

$\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$

where, coordinates of one vertices is (A,0) and other is (0,B).where B is the point where ship is located.

C²=A²+B²

84²=A² + 40²

A²= 7056 -1600

A²=5456

A=73.86 (approx)

So, the equation of path of ship which is in the form of hyperbola is

$=\frac{x^2}{5456}-\frac{y^2}{40^2}=1\\\\=\frac{x^2}{5456}-\frac{y^2}{1600}=1$