Write the equation of the hyperbola with Focus at (2,0). What is the equation of its directrix?

Answer 1

To solve this problem it must be assumed that the center of hyperbola is at the origin,

Since the given focal point is to the right of the origin , the equation of the hyperbola is of the form:

(x/a)² - (y/b)² = 1

The distance of the focal point from the center is 2 = sqrt( a² + b²) . The simplest case is a = b = sqrt(2)

So the equation is then x² / 2 - y² / 2 = 1

The directrix is a vertical line located a² /( a² + b²) = 1/2 from the origin so x = 1/2.

There is actually a whole family of solutions with a and b obeying 2 = sqrt(a² + b²)

If the question were about a parabola, the form of the equation would be x = a y²

The vertex is the point ( 0, 0) . The distance from the vertex to the focal point is 2. The distance from the vertex to the focal point is the P value. The P value obeys the relation P = 1/(4a) . So 1/(4a) = 2 ; a = 8

The directrix is a vertical line located at the distance P from the vertex. In this case that is a line located 2 to the left of (0,0) . The equation of the directrix is thus x = - 2