Find the focus and Directrix of the parabola x+y^2=0

Question

I have been able to do every answer but this about focus and directix, can someone help solve and help me understand what's the difference when y has the exponent instead of the x?

Answer 1

Here's the simplest way to look at it.

If the equation can be written y = ax^2 with a being positive then the parabola opens upward.

If the equation can be written y = ax^2 with a being negative then the parabola opens downward.

If the equation can be written x = ay^2 with a being positive then the parabola opens to the right.

If the equation can be written x = ay^2 with a being negative then the parabola opens to the left.

This equation falls into the last category and can be written y^2 = -x so it opens to the left.

Now lets put the equation into standard form (y - k)^2 = 4p(x - h)

(y - 0)^2 = -1(x - 0)

h and k are both zero here so the vertex is(0,0) which is the origin.

p is the focal distance and since 4p = -1 the p = -1/4.

Since the parabola opens to the left the focus will be 1/4 units to the left of (0,0) at (-1/4,0). (The focus is always 'inside' the parabola.)

The directrix will be a vertical line 1/4 units to the right of (0,0). Its equation will be x = 1/4.

Answer 2

Answer 3

For a vertical parabola (x has the exponent) we use the generic formula: (x - h)² = 4p(y - k)

For a horizontal parabola (y has the exponent instead of x) we use basically the same formula but with the stuff inside the parenthesis switched around: (y - k)² = 4p(x - h)

In both formulas, p is the distance from the vertex to the focus and it is also the distance from the vertex to the directrix (but on the other side of the vertex). The focus will always sit inside the parabola while the directrix will be a line behind the parabola. Recall that the vertex in both formulas is (h, k).

The equation we are given, x + y² = 0, doesn't quite match the formula above so we need to make some rearrangements:

(1) x and y need to be on opposite sides of the equation so let's move x to the other side. We do this by subtracting x from both sides. This gives us: y² = -x

(2) Next we need to figure out h and k. In this case there are no other numbers in the original equation so both h and k are 0. To see this we can add some parenthesis and see that nothing changes in the equation: -(x - 0) is the same as -x, and (y - 0)² is the same as (y²). Now that we know the values for h and k we can write our formula as: (y - 0)² = -(x - 0)

(3) Finally we need to what to do about the 4p that is supposed to be in front of the (x - 0) term. To do this we need to ask ourselves what we can set p to so that our formula, (y - 0)² = -(x - 0), matches the generic formula, (y - k)² = 4p(x - h). Well, there is no number (coefficient) in front of our (x - 0) term, just a negative sign, which always means that the coefficient is -1, similarly, if there was no number or negative sign the coefficient would be 1. We can see that -(x - h) is the same as -1(x - h), so let's rewrite out formula:

(y - 0)² = -1(x - 0).

Almost done! comparing our formula with the generic formula we can see that h and k are both zeros (from step 2) and 4p is sitting in the place where we have a -1. This means 4p has to be equal to -1. Let's solve this equality to find p.

(3.1) Our starting equation: 4p = -1

(3.2) Divide both sides by 4 to get: p = -1/4 (in fraction form) or p = -0.25 (in decimal form)

(4) Now we have h = 0, k = 0 and p = -1/4, this is everything we need! The vertex is (h, k) so our vertex is (0, 0) and the distance from the vertex to the vertex is -1/4. Keep in mind, p is negative we will be moving in a negative direction away from the vertex to find the focus. Since out parabola is horizontal we will be moving right or left to find the focus.

(4.1) Focus: From our vertex (0, 0) we will move 1/4 to the left (negative direction), this gives us a focus of (-1/4, 0)

(4.2) Directrix: The directrix will be the same distance from the vertex but in the opposite direction so we will move 1/4 to the right, this puts the directrix at (1/4, 0), but remember that the directrix is a line not a point like the focus. Since our parabola is horizontal the directrix will be a vertical line (remember all vertical lines have the equation x = ?, where the question mark is the place on the x-axis that the line crosses through). In our case the directrix is at the x location of 1/4 (from the coordinate (1/4, 0) that we found) so our directrix has the equation x = 1/4.