Question

a parabola is represented by the equation ( y^2 = 5x ). which equation represents the directrix? ( \bigcirc y = -20 ) ( \bigcirc x = -20 ) ( \bigcirc y = -\frac{5}{4} ) ( \bigcirc x = -\frac{5}{4} )

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens to the right or left is \( y^2 = 4ax \), where the vertex is at the origin \((0,0)\), the focus is at \((a,0)\), and the directrix is the line \( x=-a \).

Step2: Compare with the given equation

The given equation is \( y^2 = 5x \). Comparing it with \( y^2 = 4ax \), we have \( 4a = 5 \).

Step3: Solve for \( a \)

To find \( a \), we divide both sides of \( 4a = 5 \) by 4: \( a=\frac{5}{4} \).

Step4: Find the directrix

For the parabola \( y^2 = 4ax \), the directrix is \( x=-a \). Substituting \( a = \frac{5}{4} \), we get the directrix as \( x = -\frac{5}{4} \).

Answer:

\( x = -\frac{5}{4} \) (corresponding to the option: \( x = -\frac{5}{4} \))