IIT JEE diameter slope

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.