Question

find the focus, directrix, focal diameter, vertex and axis of symmetry for the parabola 42.4x = y²
focus =
directrix =
focal diameter =
vertex =
axis of symmetry =
be sure to enter each answer in the appropriate format. hint: what is the appropriate notation for a line or a point?
question help: video

Explanation:

Step1: Rewrite the equation in standard form

The standard form of a parabola of the form \(y^{2} = 4px\) is used. Given \(42.4x=y^{2}\), we can rewrite it as \(y^{2}=4\times10.6x\).

Step2: Find the focus

For the parabola \(y^{2}=4px\), the focus is \((p,0)\). Here \(p = 10.6\), so the focus is \((10.6,0)\).

Step3: Find the directrix

The directrix of the parabola \(y^{2}=4px\) is \(x=-p\). Substituting \(p = 10.6\), the directrix is \(x=- 10.6\).

Step4: Find the focal diameter

The focal diameter of the parabola \(y^{2}=4px\) is \(|4p|\). Since \(p = 10.6\), the focal diameter is \(|4\times10.6|=42.4\).

Step5: Find the vertex

The vertex of the parabola \(y^{2}=4px\) is \((0,0)\).

Step6: Find the axis of symmetry

For the parabola \(y^{2}=4px\), the axis of symmetry is \(y = 0\) (the \(x -\)axis).

Answer:

Focus: \((10.6,0)\)
Directrix: \(x=-10.6\)
Focal diameter: \(42.4\)
Vertex: \((0,0)\)
Axis of symmetry: \(y = 0\)