Challenge problem: Points on two circles

Point A is at negative 5 comma 5. So this is negative 5 right over here. This is 1, 2, 3, 4, 5. That's 5 right over there. So point A is right about there. So that is point A, just like that, at negative 5 comma 5. And then, it's a center of circle A, which I won't draw just yet, because I don't know the radius of circle A. Point B is at-- let me underline these in the appropriate color-- point B is at 3 comma 1. So 1, 2, 3 comma 1. So that's point B right over there, so center of circle B. Point P is at 0, 0, so it's right over there at the origin. And it is on circles A and B. Well, that's a big piece of information. Because that tells us, if this is on both circles, then that means that this is B's radius away from the point B from the center. And this tells us that it is circle A's radius away from its center, which is at point A. So let's figure out what those radii actually are. And so we can imagine-- let me draw a radius or the radius for circle A-- we now know since P sits on it, that this could be considered the radius for circle A. And you could use a distance formula, but what we'll see is that the distance formula is really just falling out of the Pythagorean theorem. So the distance formula tells us the radius right over here, this is just the distance between those two points. So the radius or the distance between those two points squared is going to be equal to our change in x values between A and P. So our change in x values, we could write it as negative 5 minus 0 squared. That's our change in x, negative 5 minus 0 squared, plus our change in y, 5 minus 0 squared, which gets us that our distance between these two points, which is the length of the radius squared, is equal to negative 5 squared plus 5 squared. Or we could say that the radius is equal to the square root of, this is 25, this is 25, 50. 50, we can write as 25 times 2. So this is equal to the square root of 25 times the square root of 2, which is 5 times the square root of 2. So this distance right over here is 5 times the square root of 2. Now, I said this is just the same thing as the Pythagorean theorem. Why? Well, if we were to construct a right triangle right over here, then we can look at this distance. This distance would be the absolute value of negative 5 minus 0. Or you could say, this is 0 minus negative 5. This distance right over here is 5. This distance is the distance between 0 and 5 in the y direction. That's 5. Pythagorean theorem tells us that 5 squared, which is 25, plus 5 squared, another 25, is going to be equal to your hypotenuse squared. And that's exactly what we have here. Now, you might be saying, wait, wait, wait, this thing had a negative 5 squared here, while here, you had a positive 5. But the reason why we could do this is when you square it, the negative disappears. The distance formula, you could write it this way, where you're taking the absolute value. And then, it becomes very clear that this really is just the Pythagorean theorem. This would be 5 squared plus 5 squared. 5 squared plus 5 squared. The reason why you don't have to do this is because a sign doesn't matter when you square it. It always will become a positive value. But either way, we figured out this radius. Now, let's figure out the radius of circle B. Same exact idea. The radius of circle B squared is equal to our change in x. So we could write as 3 minus 0, or 0 minus 3. But we'll just write it as 3 minus 0. 3 minus 0 squared plus 1 minus 0 squared. Or the radius or the distance between these two points is equal to the square root of-- let's see, this is 3 squared plus 1 squared. That's 9 plus 1. This is the square root of 10. The radius of B is the square root of 10. Now, they ask us, which of the following points are on circle A, circle B, or both circles? So all we have to do now is look at these points. If this point is the square root of 10 away from point B, then it's on the circle. It's a radius away. A circle is the locus of all points that are a radius away from the center. If it's 5 square roots of 2 from this point, then it's on circle A. If it's neither, then it's neither. Or it could be both. So let's try these out, one by one. So point C is at 4, negative 2. So let me color this in a new color. So point C-- let me do it in orange-- point C is at 1, 2, 3, 4, negative 2. Point C is right over there. Now, it looks pretty close. Just, this is a hand-drawn drawing, so it's not perfect. So point C is there. It looks pretty close. But let's actually verify it. The distance between point C and point D, so the distance squared, is going to be equal to the change in x's. So we could say, 4 minus-- so we're trying [INAUDIBLE] C and B, is 4 minus 3 squared plus negative 2 minus 1 squared, which is equal to-- This is 1 squared plus negative 3 squared. And so our distance squared is equal to 10. Or our distance is equal to the square root of 10. So this is also, the distance right over here is the square root of 10. So this is on the circle. If we wanted to draw circle B, it would look something like this. And once again, I'm hand-drawing it, so it's not perfect. But it would look something-- I'm going to draw part of it-- it looks something like this. This is exactly a radius away. So let me write, this is on circle B. Now, let's look at this point. The point 5 comma 3. So I'll do that in pink. So 1, 2, 3, 4, 5 comma 3. So this looks close, but let's verify, just in case. So now, our distance is equal to-- let me just write it this way-- our distance squared is going to be our change in x squared. So 5 minus 3 squared plus 3 minus 1 squared, change in y. And so our distance is going to be equal to-- actually I don't want to skip too many steps. Let's see, this is 2 squared, which is 4, plus 2 squared, which is another 4. So our distance is going to be equal to the square root of 8, which is the same thing as the square root of 2 times 4, which is the same thing as 2 times the square root of 2. Square root of 4 is 2. And then, of course, you just have the 2 left in the radical. So this is a different distance away than square root of 10. So this one right over here is definitely not on circle B. And just eyeballing it, you can see that it's not going to be on circle A. This distance, just eyeballing it, is much further than 5 square roots of 2. And that's also true for point C. Point C is much further than 5 square roots of 2. You can just look at that visually. They're much further than a radius away from A. So this point right over here, this is neither. This is on neither circle. Now, finally, we have the point, negative 2 comma 8. So let me find-- I'm running out of colors. Let me see, I could use, so I guess I'll use yellow again. Negative 2 comma 8, so that's negative 2 comma 1, 2, 3, 4, 5, 6, 7, 8. So it's right over here. That is point E. Just eyeballing it, this distance-- so it's clearly way too far. Just looking at, just eyeballing it, it's clearly more than a radius away from B. So this isn't going to be on circle B. And also, looking at it relative to point A, it looks much closer to point A. It doesn't even seem close, than point P is. So it looks, just inspecting it, that you could rule this one out, that this is going to be neither. But we can verify this on our own, if we like. We can just find the distance between these two points. Our distance squared is going to be our change in x's. So negative 2 minus negative 5 squared plus our change in y. So it's 8 minus 5 squared. And so this is, our distance squared is going to be equal to negative 2 minus negative 5. That's negative 2 plus 5. So that's going to be 3 squared plus 3 squared. And you see that right over here, Pythagorean theorem. This distance right over here is 3. This distance right over here, this is your change in x is 3. Change in y is 3. 3 squared plus 3 squared is going to be the distance squared, the hypotenuse squared. So our distance squared is going to be-- or I could say, our distance-- skip a few steps-- is equal to the square root of-- we can write this as 9 times 2 or the distance is equal to 3 times the square root of 2. The radius of circle A is 5 times the square root of 2, not 3 times the square root of 2. So this is actually going to be inside the circle. So if we want to draw circle A, it's going to look something like this. And point E is on the inside. Point D and point C are on the outside of circle A. The only one that sits on any of the circles is point C.