Question

find the focus, directrix, focal diameter, vertex and axis of symmetry for the parabola 35.2y = x²
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: Write the equation in standard form

The standard form of a parabola that opens up or down is \(x^{2}=4py\).
Given \(35.2y = x^{2}\), we can rewrite it as \(x^{2}=35.2y\). Comparing with \(x^{2}=4py\), we have \(4p = 35.2\), so \(p=\frac{35.2}{4}=8.8\).

Step2: Find the focus

For a parabola \(x^{2}=4py\) that opens up, the focus is at the point \((0,p)\).
Since \(p = 8.8\), the focus is \((0,8.8)\).

Step3: Find the directrix

The directrix of a parabola \(x^{2}=4py\) is the line \(y=-p\).
Since \(p = 8.8\), the directrix is \(y=-8.8\).

Step4: Find the focal diameter

The focal diameter of a parabola \(x^{2}=4py\) is \(|4p|\).
Since \(4p = 35.2\), the focal diameter is \(35.2\).

Step5: Find the vertex

For a parabola of the form \(x^{2}=4py\), the vertex is at the point \((0,0)\).

Step6: Find the axis of symmetry

For a parabola of the form \(x^{2}=4py\), the axis of symmetry is the \(y -\)axis. The equation of the \(y -\)axis is \(x = 0\).

Answer:

Focus = \((0,8.8)\)
Directrix = \(y=-8.8\)
Focal diameter = \(35.2\)
Vertex = \((0,0)\)
Axis of symmetry = \(x = 0\)