Problem 14 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(x^2 = 6y\).

Step 2: 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 \(a = 6/4 = 1.5\).

Step 3: 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 \((0, 1.5)\).

Step 4: 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 \(y = -1.5\).

Step 5: 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.