Question

find the focus, directrix, focal diameter, vertex and axis of symmetry for the parabola 24.4y = 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: Rewrite the equation in standard form

The standard form of a parabola that opens up or down is \(x^{2} = 4py\). Given \(24.4y=x^{2}\), we can rewrite it as \(x^{2}=24.4y\). Comparing with \(x^{2} = 4py\), we have \(4p=24.4\), so \(p=\frac{24.4}{4}=6.1\).

Step2: Find the focus

For a parabola \(x^{2}=4py\) that opens up (since \(p = 6.1>0\)), the focus is at the point \((0,p)\). Substituting \(p = 6.1\), the focus is \((0,6.1)\).

Step3: Find the directrix

The directrix of a parabola \(x^{2}=4py\) is the line \(y=-p\). Substituting \(p = 6.1\), the directrix is \(y=-6.1\).

Step4: Find the focal diameter

The focal diameter of a parabola \(x^{2}=4py\) is \(|4p|\). Since \(4p = 24.4\), the focal diameter is \(24.4\).

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

Answer:

Focus = \((0,6.1)\)
Directrix = \(y=-6.1\)
Focal diameter = \(24.4\)
Vertex = \((0,0)\)
Axis of symmetry = \(x = 0\)