Question

find the vertex, focus, and directrix for the parabola \\(y^2 = -20x\\)

vertex:

focus:

equation of the directrix:

Explanation:

Step1: Identify the standard form

The given equation \(y^2 = -20x\) is a horizontal parabola.

$$y^2 = 4px$$

Step2: Determine the vertex

Since there are no translation terms, the vertex is at the origin.

$$(h, k) = (0, 0)$$

Step3: Calculate the value of p

Equate the coefficients of \(x\) to find \(p\).

$$4p = -20 \implies p = -5$$

Step4: Find the focus

The focus of a horizontal parabola is \((h + p, k)\).

$$(0 + (-5), 0) = (-5, 0)$$

Step5: Find the directrix

The equation of the directrix is \(x = h - p\).

$$x = 0 - (-5) \implies x = 5$$

Answer:

Vertex: \((0, 0)\)
Focus: \((-5, 0)\)
Equation of the directrix: \(x = 5\)