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Current time:0:00Total duration:7:48

Let A and B be two distinct
points on the parabola y squared is equal to 4x. If the axis of the parabola
touches a circle of radius r, having AB as its
diameter, then the slope of the line joining
A and B can be. So let's draw this. So let me draw my axes here. That's my y-axis. And here is my x-axis. And then the parabola y
squared is equal to 4x would look like this. Something like that. That would be in
the first quadrant, and then it would be
symmetric around that. It would look like that. So this would be y
squared is equal to 4x. Or if I just focused on
this top part over here, I could take the principal
root of both sides. And I get y is equal to 2
times the square root of x. Just taking the square
root of both sides, the square root of 4x is 2
times the square root of x. This down here would be y
is equal to negative 2 times the square root of x. So the positive square root,
the negative square root of both sides. But this one defines this
whole parabola implicitly. And then they say, they
tell us that if the axis of the parabola--
that's essentially the x-axis in this drawing--
if the axis of the parabola touches a circle of radius r
having AB as its diameter-- So let's take two
points on this parabola. So let's say that
that is point A, and let's say that
this is point B. It is the diameter
for some circle. So that's the diameter
for some circle that just touches this axis. It'll just touch on here. So let me see how well
I can draw that circle. Obviously, this will
not be an exact drawing. But let me try my best to draw
the circle under question. So the circle will
look-- I can do a better job than that-- the circle
will look like that. That's my best shot at it. And then it has a radius r. So if we say the
center of the circle, and obviously the
diameter will go through the center
of the circle, this distance right here is r. So another way to think
about it, this y value right over here,
that y value is r. Now, let's think a little
bit about the center of this circle. Because maybe with the
center of the circle, we can relate the A's
and the B's to the r's. So let me define A and B. So
let me define the point A. Let's say the point A,
just for convenience, let's make x is equal
to lowercase a squared. If x is lowercase a squared,
then the square root of that is a times 2. Then y will be 2a. So this right here is
the point a squared 2a. And let's make B the point
using the same logic. Let's just say that's
lowercase b squared. And I'm doing that so that
when I take the square root, it'll become simple. So the square root of b
squared is b times 2 is 2b. Now, let's get the
coordinates for the center of that diameter. Now, the center of
that diameter is just the midpoint of this
point and that point. So let me write it down here. The midpoint of
the points A and B is going to be equal to the
average of the x values, a squared plus b squared over
2, and the average of the y values. So this will be 2a plus
2b, all of that over 2. And of course the 2s cancel out. And so this is going
to be a plus b. That is the y value
of the midpoint. So this point right
over here is going to be a squared plus b
squared over 2 comma a plus b. Now, we just saw that this
height right here is r. So a plus b, this y value right
over here is also equal to r. This is equal to a plus b. Write the y value in
between 2a and the y value in between 2b right over here. 2b. This is a plus b. So just like that,
we were able to get a relationship
between a, b, and r. So we know a plus
b is equal to r. Now, let's try to find the
actual slope of this diameter. That's what they want to ask us. The slope of the line
joining a and b can be. So let's figure out the
slope of this actual line, and let's see if
we can relate it to r in some way maybe
using this information. Because all the answers they
give us are in terms of r. So we need to have an
answer in terms of r. So the slope is going to be
change in y over change in x. So 2a minus 2b over a
squared minus b squared. This is equal to 2 times
a minus b over a plus b-- this is just a difference
of squares-- a plus b times a minus b. The a minus b's cancel out. The slope of the
diameter, the slope of the line that we care about
is going to be 2 over a plus b. It's 2 over a plus b. Now, that's clearly
not one of the answers. We need our answer
in terms of r. But we just figured
out the y-coordinate of the midpoint,
which was a plus b. That is equal to r. So this is the
same thing as 2/r. So the slope of this line
right over here is 2/r. Now, you might be tempted
to say OK, the answer is C. And I actually probably
should have told you this at the beginning
of this video in case you wanted to pause it
and try it yourself, is that this is actually
a multiple correct answer problem. So you could actually have
multiple choices here. So we want to make sure that
this isn't the only answer. And it's not the only answer. Because we could have put
a here and b here and drawn the circle like this. Essentially the
exact same circle, but in the fourth quadrant. So I could have drawn
the circle like this. And then I would
have been dealing with the slope of this
diameter would be in question. And you can see
everything is symmetric. It's just flipped around
the axis of symmetry. It's flipped around the x-axis. Whatever is the slope up
here, the negative of it is going to be the
slope over here. So this is also a
potential slope. So if this is 2/r, then
this slope over here is negative 2/r. So this is also an option. And just based on the
reasoning we just did, you could have ruled
out two of these. If you knew that
this slope is a, or if you know that this
slope up here is going to be, I don't know, if this slope
up here was going to be r, then the slope over here
is going to be negative r. If the slope up here is
going to be a plus b, this slope would be
negative a plus b. So you know that you would
have to have two answers that are the negative of each other. So when you look at just the
choices, you could have said, oh, it's either going to
be A and B because they're the negatives of each other,
or it's going to be C and D. So you could have
ruled out-- you could have either just picked
these two or these two. So you had a 50/50
chance of guessing. But we don't want to guess. We want to solve it. And we just did. Anyway, hopefully
you enjoyed that.