Question

find the equation for the parabola that has its vertex at the origin and has directrix at ( x=\frac{1}{43} ). the equation is: question help: video written example

Explanation:

Step1: Identify parabola orientation

Directrix \( z = \frac{1}{43} \) is horizontal, so parabola opens along z-axis.

Step2: Recall standard form

For vertex at origin, directrix \( z = -p \) (wait: correction—directrix \( z = k \), vertex at (0,0), so if directrix is \( z = \frac{1}{43} \), the focus is at \( (0,0,-\frac{1}{43}) \), so \( p = -\frac{1}{43} \)? No, standard form: distance from point (x,y,z) to vertex focus equals distance to directrix. Let focus be (0,0,-c), directrix \( z = c \), so equation is \( x^2 + y^2 = -4cz \). Here directrix \( z = \frac{1}{43} \), so \( c = \frac{1}{43} \).

Step3: Calculate 4c

\( 4c = 4 \times \frac{1}{43} = \frac{4}{43} \)

Step4: Write equation

\( x^2 + y^2 = -4cz = -\frac{4}{43}z \)

Answer:

\( x^2 + y^2 = -\frac{4}{43}z \)