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Chapter 7: Problem 14

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$x^{2}-6 y=0$$

Short Answer

Expert verified
The focus of the parabola is at (0, 1.5) and the equation of the directrix is \(y = -1.5\) .

Step by step solution

01

Express the given equation in the form of \(x^2 = 4ay\)

Express the given equation \(x^2 - 6y = 0\) in the form of \(x^2 = 4ay\). Rearranging terms, we get \(x^2 = 6y\).
02

Identify the value of \(a\)

In step 1, we got \(x^2 = 6y\) which is of the form \(x^2 = 4ay\). Comparing both equations, we can conclude that \(4a = 6\). Solving for \(a\), we get \(a = 6/4 = 1.5\).
03

Find the focus of the parabola

Given that the formula for the focus of a parabola \(x^2 = 4ay\) is \((0, a)\). Substituting \(a = 1.5\) in the equation, we get focus as \((0, 1.5)\).
04

Find the directrix of the parabola

The equation of the directrix for a parabola \(x^2 = 4ay\) is \(y = -a\). Substituting \(a = 1.5\) in the equation, we get directrix as \(y = -1.5\).
05

Graph the parabola

Plot the parabola using the focus (0, 1.5) and the directrix \(y = -1.5\) . The vertex of the parabola is at the origin (0,0), the parabola opens upwards. The y-coordinate for the vertex, focus, and root of the directrix are in a straight line.

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