Question

find the coordinates of the focus and equation of the directrix for the parabola given by $y^2 = -4x$. the general formula for this parabola is $y^2 = 4px$. therefore, the value of $p$ is dropdown. the coordinates of the focus dropdown. the equation of the directrix dropdown with -1, 1, 4.

Explanation:

Step1: Compare with general formula

The general formula is \( y^2 = 4px \), and our equation is \( y^2=-4x \). So we set \( 4p=-4 \).

Step2: Solve for \( p \)

Divide both sides of \( 4p = - 4 \) by 4: \( p=\frac{-4}{4}=-1 \).

Step3: Find focus coordinates

For a parabola \( y^2 = 4px \), the focus is at \( (p,0) \). Since \( p = - 1 \), the focus is \( (-1,0) \).

Step4: Find directrix equation

The directrix of \( y^2 = 4px \) is \( x=-p \). Since \( p=-1 \), then \( -p = 1 \), so the directrix is \( x = 1 \).

Answer:

The value of \( p \) is \(-1\). The coordinates of the focus are \((-1,0)\). The equation of the directrix is \( x = 1 \).