Derive the equation of the parabola with a focus at (6,2) and a directrix of y=1

If the focus is at (6, 2) and the directrix is a horizontal line y = 1, then that tells us that is an x^2 parabola.  Since the parabola hugs the focus, it will open upwards since the focus is above the directrix.  The rule here is that the vertex is the same distance from the focus as it is from the directrix.  If the focus is at a y-value of 2 and the directrix is at y = 1, then the vertex is right in between them as far as the y coordinate goes, which is 1.5.  It will have the same x coordinate at the focus.  The vertex is in the form (h, k), so our h is 6, and our k is 1.5.  The vertex is (6, 1.5).  The standard form of a parabola of this type is , where p is the distance from the vertex to the focus.  Our p is 1/2.  Using the h and k from the vertex, and the p of 1/2, we now have this as our equation, not yet simplified: .  That will simplify a bit to .  Depending upon how you are to state your answer, how it needs to "look" in the end will vary.  I am going to FOIL the left and distribute the right and then put everything on one side and set it equal to y.  That would be this: .  And there you go!

put like terms together.orremember that the 45 becomes negative when moved to the other side.orthe minuses cancel out.

38+105= 143, 39+104=143, and you will find then adding the numbers on opposite sides, moving inwards yields the answer 138 every time. because of this, you’re going to multiply 138 times the number of pairs, which is 34. therefore, 34*138= 4692(i’m sorry if it’s the wrong numbers, but it’s the right concept)