Equation of the parabola: y² = 6x
Coordinates of the focus: (1.5, 0)
Equation of the directrix: x = -1.5
Length of the straight side: 6 units
Finding the Equation of the Parabola:
Since the parabola has its vertex at the origin and its axis of symmetry coincides with the positive x-axis, its equation is of the form:
y² = 4ax
where "a" is a constant that determines the shape of the parabola.
We are given that the parabola passes through the point (6, 6). Substituting these values into the equation, we get:
6² = 4a(6)
36 = 24a
a = 1.5
Therefore, the equation of the parabola is:
y² = 6x
Finding the Coordinates of the Focus:
The focus of a parabola is always located a distance of "a" units from the vertex along the axis of symmetry. In this case, "a" is 1.5, so the focus is located at (1.5, 0).
Finding the Equation of the Directrix:
The directrix of a parabola is always located a distance of "a" units from the vertex in the perpendicular direction to the axis of symmetry. In this case, the axis of symmetry is horizontal, so the directrix is a vertical line located 1.5 units to the left of the vertex. Therefore, the equation of the directrix is:
x = -1.5
Finding the Length of the Straight Side (Latum Rectum):
The latus rectum of a parabola is the segment that passes through the focus and is perpendicular to the axis of symmetry. Its length is equal to 4 times the distance from the focus to the vertex, which is 4 * 1.5 = 6.
Therefore, the length of the latus rectum is 6 units.
Summary:
Equation of the parabola: y² = 6x
Coordinates of the focus: (1.5, 0)
Equation of the directrix: x = -1.5
Length of the straight side: 6 units
Complete question:
Find the equation of the parabola, the coordinates of its focus, the equation of the directrix and the length of its straight side. Say that the vertex is at the origin and whose edge of symmetry coincides with the x edge passing through the point (6,6).