Question

HELP!!! show that the circle having the latus rectum of a parabola as diameter is tangent to the directrix.

Answer 1

hello again

Answer 2

hello!! yeheey :)

Answer 3

lets put the parabola with vertex at the origin, and write its equation as \(x^2=4py\)

Answer 4

therefore the length of the latus rectum is \(4p\) right?

Answer 5

yesyes

Answer 6

so if that is the diameter of the circle, then the radius of the circle is half of that, namely \(2p\)

Answer 7

oh damn damn damn i screwed it up again
jeez
but it is close

Answer 8

replace all the \(2p\) by just \(p\)

Answer 9

it would be a lot easier if i could draw it
let me try

Answer 10

the center of the circle is \((0,p)\) and the radius of the circle is \(2p\)

Answer 11

from the center \((0,p)\) to the directrix \(y=-p\) is exactly \(2p\) units away, which is the radius of the circle
so it must touch the directrix at the point \((0,-p)\)