If the area of an equilateral triangle inscribed in a parabola is `6912sqrt3` , then what is the length of the latus rectum of the parabola?
First, let's break down the triangle. Because its area is `6912sqrt3` we can determine its base length and height. Recognize that in an equilateral triangle, the height will be the following (given a side length of `s`):
`h = ssqrt3/2`
So, our area becomes:
`A = 1/2 bh = 1/2*s*ssqrt3/2 = s^2sqrt3/4`
So, we solve for a side length of the triangle:
`6912sqrt3 = s^2sqrt3/4`
Now, multiply by `4/sqrt3`:
`27648 = s^2`
Now, take the square root of both sides:
`96sqrt3 = s`
Alright! Now, we have a side length. Now, we can begin to define our parabola. For simplicity, let's let the focus of the parabola lie at the origin and have it open upward. We then also know that the three points that define the equilateral triangle are one right at the vertex and two symmetricall across the y axis (the triangle will be upside down). Let's let the triangle be defined by points A (at the vertex of the parabola), B, and C and using the side length we found above, we can determine the following.
`B = (-48sqrt3, a+144)`
How did I get those last two points? Recall that the height will give the difference in y-coordinates between the apex and base of the triangle. The horizontal coordinate will be found by dividing the side length by 2 because of the triangle's symmetry.
We now know that the directrix will be twice the distance away from the focus as the vertex, giving the equation of the directrix:
`y_d = 2a`
Alright, now, when calculating the length of the latus rectum, we just have to recognize that it has a simple relation to the length of the chord between the focus and directrix: it's double that length! Keep in mind that sometimes the way that the directrix or focus is defined, `a` may be negative, so we need to take the absolute value:
`L_(lr) = |4a|`
Ok, so, here's the hard part. What is a? Well, we know that that each point on the parabola has an equivalent distance to the focus and to the directrix. So, for any (x,y), the following must hold:
`y-2a = sqrt(x^2+y^2)`
The left side gives us the distance from the point (x,y) to the directrix, and the right side uses the distance formula to compute the distance from (x,y) to the focus (0,0). Let's try to simplify:
`y^2-4ay+4a^2 = x^2+y^2`
We can subtract `y^2` from both sides:
`-4ay+4a^2 = x^2`
`4a(a-y) = x^2`
Now, to solve for a, let's plug in the points B or C (they give the same result because `x^2 = (-x)^2`
`4a(a-(a+144)) = (48sqrt3)^2`
Now, we can divide both sides by -144:
`4a = -48`
And there we have our answer: 48 (remember how above we figured out that `L_(lr) = |4a|`). We have a negative distance because of how we defined the parabola. However, using the formula we found:
`L_(lr) = 48`
Hope that helps!