Question

What is the equation for the parabola with focus (4, 1) and directrix x = 2?

Option 1) 4x + 1 = (y – 12)²
Option 2) 4x + 12 = (y - 1)
Option 3) 4x - 1 = (y - 12)
Option 4) 4x - 12 = (y - 1)

Answer 1

The equation for the parabola with focus (4,1) and directrix x=2 is Option 4) 4x - 12 = (y - 1), derived using the definition of a parabola as a set of points equidistant from the focus and the directrix.

Step-by-step explanation:

The equation for the parabola with focus (4,1) and directrix x=2 can be derived using the definition of a parabola as the set of all points that are equidistant from the focus and the directrix. For any point (x,y) on the parabola, the distance to the focus (4,1) is equal to the distance to the directrix (x=2).

Thus, the equation is
(x-4)^2 + (y-1)^2 = (x-2)^2.

Expanding both sides and simplifying, we get

(x^2 - 8x + 16) + (y^2 - 2y + 1) = (x^2 - 4x + 4)

Removing the quadratic terms for x and simplifying further, the equation

4x - 12 = (y - 1)^2

emerges, which corresponds to Option 4) 4x - 12 = (y - 1).