Question 892102: given the (2,-2) and (2,10) are the ends of latus rectum of a certain parabola. What is the distance between the vertex?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the latus rectum is the line perpendicular to the axis of symmetry that goes through the focus of the parabola.
the length of the latus rectum is equal to 4p
your latus rectum is a vertical line which means that the major axis of your parabola is horizontal.
since the axis of symmetry is the line in between the points (2,-2) and (2,12), then your axis of symmetry is the line y = 4 because 4 is halfway between -2 and 12,
the equation to find the focus of a parabola that is horizontally aligned is:
4p * (x-h) = (y-k)^2
since 4p is the distance between the points on the latus rectum, then 4p must be equal to 12.
your equation becomes 12 * (x-h) = (y-k)^2
now p is the distance between the focus and the vertex, and p is also the distance between the directrix and the vertex.
this is because the vertex is halfway between the focus and the directrix on the axis of symmetry.
this places the vertex at (-1,4) because p = 3.
the directrix is the line at x = -4 which is 3 units from the vertex on the other side of the vertex.
the major property of a parabola is that the distance between the focus and a point on the parabola is the same as the difference from that point to the directrix.
since the distance between the focus and a point on the parabola is equal to 6, this means that the distance between that point and the directrix is also 6.
since the vertex is halfway between the focus and the directrix, then the distance to the vertex has to be half of that which is 3.
we already knew that from our previous analysis, but it's nice to know that the different sets of logic confirm each other.
so the vertex of the parabola is equal to (-1,4).
the formula of 4p(x-h) = (y-k)^2 becomes 12(x+1) = (y-4)^2
if we graph this equation, we should be able to see our parabola.
here's a graph of your parabola.
see below the graph for further comments.
the equation that was graphed was the equation of 12(x+1) = (y-4)^2
to graph this equation, you would have to solve for y to get y = 4 +/- sqrt(12*(x+1))
if you solve that equation for x, you will get x = (y^2 - 8y + 4) / 12
if you knew what you were doing up front, you would have known that the length between the two points on the latus rectum is equal to 4p.
since the distance between those points is 12, then 4p must be equal to 12 and therefore p must be equal to 3.
since p is the distance between the focus and the vertex, then the vergex had to be 3 units away from the focus.
but which direction?
they didn't ask that so you didn't need to answer that.
all you needed to say was that the distance between the focus and the vertex was 3.
was it 3 left or 3 right?
could be either way as you will see shortly.
the equation may have been pointing the other way.
in that case, the equation would have been 12(x-5) = -(y-4)^2
the graph of that equation is shown below:
your solution is that the distance between the vertex and the focus is 3.
this is the same as the distance between the vertex and the latus rectum since the focus is on the latus rectum.
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