Question

A parabola can be represented by the equation x2 = 2y. What are the coordinates of the focus and the equation of the directrix? focus: (0,8); directrix: y = –8 focus: (0, one-half); directrix: y = negative one-half focus: (8,0); directrix: x = –8 focus: (one-half, 0); directrix: x = negative one-half

Answer 1

Answer:

b)

i) Focus ( 0,a) = $(0,\frac{1}{2} )$

ii) The equation of the directrix is

                            $y = -\frac{1}{2} or y +\frac{1}{2} =0$

Step-by-step explanation:

Step(i):-

  A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not in a line) in the plane

• The Fixed line is called the directrix of the parabola.

• The Fixed point is called the focus of the parabola.

• A line through the focus and perpendicular to the directrix is called the axis of the parabola.

• The point of intersection of parabola with the axis is called the vertex of the parabola.

Given Parabola x² = 2 y

Comparing  x² = 4 a y

                   4 a = 2

                     $a = \frac{1}{2}$

  Focus ( 0,a) = $(0,\frac{1}{2} )$

Step(ii)

The equation of the directrix is y = -a or y +a=0

The equation of the directrix is

                            $y = -\frac{1}{2} or y +\frac{1}{2} =0$

The equation of the directrix is 2 y +1 =0