Question

Find the standard form of the equation of the parabola with a focus at (-4, 0) and a directrix at x = 4.

y2 = -8x

16y = x2

y = negative one divided by sixteenx2

x = negative one divided by sixteeny2

Answer 1

When you're given information about the locations of the focus and directrix of a parabola, the appropriate general formulas to start with are 4py=x^2 (for a vertical parabola) and 4px=y^2 (for a horizontal parabola).

Note that 'p' is the distance from the VERTEX of the parabola to the FOCUS, OR the distance from the vertex to the directrix.

Can you determine 'p' from the info given in this problem statement?

Answer 2

Is this parabola a vertical one or a horizontal one? Hint: draw the given info and then decide how to answer this question.

Answer 3

im not sure how to draw it

Answer 4

Draw a set of coordinate axes (x- and y-axes).
Draw (mark) the focus: focus at (-4, 0)
Draw the directrix (a straight line): directrix at x = 4
Ask yourself: Where would the VERTEX of the parabola be, in relationship to the focus and the directrix?

Answer 5

so its vertical ?

Answer 6

Nice drawing. But where's your vertex?

Answer 7

where would that be

Answer 8

The vertex is exactly halfway between the focus and the directrix. Try figuring that out. Where's your vertex in this case?

Answer 9

would it be (0,0)

Answer 10

Very good. Note that the vertex is equidistant between the focus (a point) and the directrix (a line). Try again to figure out whether this parabola opens horiz. or vert. and defend your answer.

Answer 11

horizontal because their both on the x plane

Answer 12

Yes, both focus and vertex are on the x-axis (not plane).

Answer 13

What is the distance between the focus and the vertex? (It's a horiz. distance).

Answer 14

4

Answer 15

Right. So, your value for p is 4.

The formula for a horiz. para. with 'p'= the distance between focus and vertex AND vertex at the origin is

4px=y^2

Answer 16

4(4)(-4)=0^2?

Answer 17

Caution: That's for a horiz. parab. that opens to the right. The formula for a horiz. parab. that opens to the left is the same, except that you'll need to insert a (-) sign.

Answer 18

Note that when you're asked for a formula for the parabola, you need to supply one that involves both x and y. Your proposed equation involves neither, so could not be right. Try again, please.

Answer 19

Hint: From your problem statement:


y2 = -8x

16y = x2

y = negative one divided by sixteenx2

x = negative one divided by sixteeny2


Which seems likeliest to be the correct answer, and why?

Answer 20

16y=x2 because i have to multiply 4 times 4

Answer 21

Think: for a horiz. parab. with vertex at (0,0), the correct equation has the form -4px=y^2, where p is the (postive) distance between vertex and focus.

Try again, please, by substituting your value for 'p' into the above equation.

Answer 22

so its the last one ?

Answer 23

@mathmale

Answer 24

Again, could you explain your reasoning and defend your answer, please?

Answer 25

-4 x 4 =-16x

Answer 26

Again, your equation must involve both x and y.

If the equation for a horiz. parab. that opens to the left is -4px=y^2, and the value of p (the positive distance between vertex and focus) is 4, what is the equation of your parabola?

Answer 27

was the answer correct

Answer 28

@Flashy32

Answer 29

@tkhunny help please

Answer 30

If you don't like the easy way, you can use the definition.

focus at (-4, 0)
directrix at x = 4.

Distance from Focus = Distance from Directrix

\(\sqrt{(x+4)^{2}+(y)^{2}} = x-4\) -- Go!