(a) Vertex: The parabola is in the form y2 =4px, where the vertex is at the origin (0,0).
Value of p: Comparing with the standard form, we have p=4.
Focus: The focus is at (p,0)=(4,0).
Focal Diameter: Focal diameter is the absolute value of 4p, so it is ∣4×4∣=16.
(b) Endpoints of Latus Rectum: Latus rectum is perpendicular to the axis of the parabola and passes through the focus. Its length is 4p, so the endpoints are (−2p,p) and (2p,p). In this case, the endpoints are (−8,4) and (8,4).
(d) Equation for Directrix: Since the parabola opens to the right, the equation for the directrix is x=−p=−4.
Axis of Symmetry: The axis of symmetry is the line x=0, passing through the vertex.
The given parabola is 4y^2 =16x.
(a) Vertex: For a parabola in the form y^2 =4px, the vertex is at the origin (0,0).
Value of p: In the given equation 4y^2 =16x, comparing with the standard form y^2 =4px, we find p=4.
Focus: The focus of the parabola is at the point (p,0)=(4,0). This is because the parabola opens to the right, and the focus is located at a distance p from the vertex along the axis of symmetry.
Focal Diameter: The focal diameter is the absolute value of 4p, which is ∣4×4∣=16.
(b) Endpoints of Latus Rectum: The latus rectum is a line segment perpendicular to the axis of the parabola and passes through the focus. Its length is 4p, so the endpoints are given by (−2p,p) and (2p,p). Substituting p=4, we get (−8,4) and (8,4).
(c) Graph: The graph of the parabola is a U-shaped curve that opens to the right. The vertex is at the origin, and the focus is to the right of the vertex.
(d) Equation for Directrix: Since the parabola opens to the right, the equation for the directrix is x=−p=−4.
Axis of Symmetry: The axis of symmetry is the vertical line passing through the vertex, which is x=0.