Question

select the correct answer from each drop-down menu.

a parabola is given by the equation (y^2 = -24x).

the equation of the directrix of the parabola is choose an answer
the focus of the parabola is choose an answer

options for the first drop-down menu:
(x = -6)
(x = 0)
(x = 6)
(y = 0)

Explanation:

🆕 New Concept Discovered: Parabolas with Horizontal Axes
How parabolas open sideways and where their focus and directrix lie.

Step 1: Identify the standard form

The given equation is:
\[ y^2 = -24x \]

This matches the standard form of a horizontal parabola centered at the origin \((0,0)\):
\[ y^2 = 4px \]

Where \(p\) is the distance from the vertex to the focus (and also from the vertex to the directrix line). Because \(y\) is squared and the coefficient of \(x\) is negative, this parabola opens to the left.

Step 2: Find the value of \(p\)

We set the coefficients of \(x\) equal to each other to solve for \(p\):
\[ 4p = -24 \]
\[ p = -6 \]

Step 3: Determine the directrix

For a horizontal parabola \(y^2 = 4px\), the directrix is a vertical line given by the equation:
\[ x = -p \]

Substituting \(p = -6\):
\[ x = -(-6) \]
\[ x = 6 \]

Step 4: Determine the focus

For a horizontal parabola centered at the origin, the focus is located at the point:
\[ (p, 0) \]

Substituting \(p = -6\):
\[ (-6, 0) \]

Answer:

• The equation of the directrix of the parabola is \(x = 6\).
• The focus of the parabola is \((-6, 0)\).