Intersection of circle & hyperbola

The circle x squared plus y squared minus 8x is equal to 0, and the hyperbola x squared over 0 minus y squared over 4 is equal to 1, intersect at the points A and B. In problem 46, they want us to find equation of the circle with AB as its diameter. So let's visualize the circle and the hyperbola. The equation of the circle x squared plus y squared minus 8x. Let me write it this way. This can be rewritten as x squared minus 8x plus y squared is equal to 0. You can add 16 to both sides. And I'm doing that to complete the square for the x term. So plus 16. This over here becomes x minus 4 squared, plus y squared is equal to 16. And so the circle's going to look something like this. That's my x-axis. That's my y-axis. Its center is at 4, 0. Zero. So 1, 2, 3, 4-- 4, 0. And its radius is 4. Its radius squared is 16. So its radius is 4. So let me go 1, 2, 3, 4-- 1, 2, 3, 4. Then we go up 4-- 1, 2, 3, 4. So all of these points, that point, that point, that point, and that point are all going to be on the circle. Let me draw that a little bit neater. That point, that point, and that point are all going to be on our circle. So my circle's going to look something like this. It's not drawn as neatly as it could be. But I think you get the general idea. This is our circle. Then they have a hyperbola. x squared over 9 minus y squared over 4 is going to equal 1. This hyperbola is going to open to the left and the right since our x term is positive. It's going to look something like this. You can actually figure out the asymptotes. Actually, let me just do that. Let me solve for y. So you would get, let me write it this way, negative y squared over 4 is equal to negative x squared over 9 plus 1. I just subtracted x squared over 9 from both sides. And then we would get, let's multiply both sides by negative 4. You get y squared is equal to 4/9. X squared minus 4. Or y is equal to-- and now this will just be the positive part over here. But I'm doing that so that we can understand its asymptotes. y is equal to the square root of 4 over 9x squared minus 4. So as x gets larger and larger and larger, this term right here is going to stop mattering so much. So as x approaches infinity, this is going to approach the square root of 4/9 x squared. This constant's not going to matter much. And so it's going to approach, this thing right over here is going to be 2/3 x. So it's going to, if you imagine a slope, let's see, 2/3. If you run 3, you rise 2. So the asymptote is going to look like that. It's going to approach that line over there. And it's going to be symmetric. So it's also going to approach-- so 3 and 2, it's also going to approach that line over there. And if we want to see where it intersects the x-axis, you just set y is equal to 0. You get x squared over 9 is equal to 1. So x is going to be equal to plus or minus 3. So the positive intersection is going to be over there. So our hyperbola on the right side's going to look something like, something, something like that. It's also going to show up on the left side. But that's less interesting because it's not doing anything with the circle. Now they told us that the hyperbola and the circle intersect at the points A and B. So this right here is the point A. This right here is the point B. In this question, we want the equation of the circle with AB as its diameter. So AB as the diameter. I could have drawn that a little bit straighter. So the equation of that circle right over there. So essentially, we just need to figure out where does this hyperbola intersect this circle. Now, the easiest way to do this is to-- well, we have two constraints here. Let's solve for y squared for the circle. And we could substitute that in for y squared right here and see what x values, the intersect that. And then we want the x value out here. So something that looks like this value is the one we want to use. Then we could figure out the y value. The y value is going to be the radius of our circle. The x value comma 0 will be the center. And then we'll have our equation. So let's do that. So this up here, we have-- let me do it in yellow. So if we subtract x squared and negative 8x from both sides, the equation of the circle can be rewritten as y squared is equal to-- I'm going to add 8x to both sides. So it's 8x. And then subtract x squared from both sides. 8x minus x squared. I really just moved to that and that over to the right hand side of the equation. Now I can take this and substitute for y squared in the equation of the hyperbola. So the hyperbola's equation is x squared over 9 minus y squared over 4 is equal to 1. Instead of running a y squared over there, we know that it has to satisfy this equation as well. So for y squared, I'm going to put an 8x minus x squared. And let's see if we can solve this right here. This is just a straight up quadratic equation, although it might not look like it just yet. Let's simplify this. So this tells us that x squared over 9 minus 8x/4. So minus 2x plus x squared over 4 plus x squared over 4. I just distributed essentially the negative 1/4 over both these terms. Is equal to 1. Let's see, we can multiply the whole thing times 36 to get rid of these fractions. So that's 4 times 9. So 36 divided by 9 is 4. So this is 4x squared minus 36 times 2 is 72. 72x plus 36 divided by 4 is 9, plus 9x squared is equal to 36. These two terms right here, we can add them. We get 13x squared, minus 72x. And then we could subtract 36 from both sides. So minus 36 is equal to 0. So now we just have a straight up quadratic equation. We just have to find the x's, find the roots. So let's use the quadratic formula here. So x is going to be equal to negative B So that's negative negative 72. So it's 72 plus or minus the square root of 72 squared. I'll just write that 72 times 72 minus 4 times A times C. C is negative 36. So that negative, you could put it out here. And just put the 36 out back. And then all of that over 2 times a. So all of that over 26. So the hard part is really to simplify this. But it looks like we have some interesting stuff going on. So let me rewrite this part over here. Let me write it over here. 72 times 72 is the same thing as 2 times 36 times 2 times 36, right? Each of those are 72. That's the same thing as 72 times 72. And then we're adding to that 4 times 36 times 13. And we're taking the square root of this whole thing. I'm just doing the work out here so we don't. waste this real estate over here. Now we can factor out a 4 times 36. This is 4 and this is a 36. So this is equal to-- I know it's getting-- let me write it a little bit neater. This is equal to the square root of, we can take a 4 times 36. And actually 4 times 36 is 144. I'll just write 4 times 36 right here. 4 times 36. Now over here we use the both 2s in the 36. So we have a 36 left. We use this 4 times 36. So we have plus 13. So this becomes the square root of, 4 times 36 is 144. 36 plus 13 is 49. So we're lucky that it actually worked out to 2 perfect squares. So this is the square root of 144 times the square root of 49, which is 12, times 7 or this is equal to 84. So this business over here simplifies to 72 plus or minus 84. This whole thing over here is simplified to 84, all of that over 26. Now if we were to subtract 84, we would get something over here that doesn't make sense in this context. We're looking for something a positive x value. So let's only consider the plus 84 and see what we get. So the first thing to do, actually let me just divide the numerator and the denominator by 2. So this is the same thing as 36 plus 42 over 13. And this is the same thing as 78 over 13. And it looks like 78 is divisible by 13. 78 divided by 13 is 6. This is equal to 6. So the x-coordinate here is 6. It's right there. So it's 6. We don't know what the y-coordinate is. So let's do that. It's pretty easy to solve. We could substitute x equals 6 into any of these equations. This one is probably easier right over here. So we get y squared is equal to 8 times 6 is 48, minus 48 minus 36, which is equal to 12. And so y is equal to the square root of 12. We could simplify that radical if we want. But we know the point now. It's 6 comma the square root-- Sorry, this point over here is 6 comma 0, but the actual intersect, the center is 6 comma 0, but the actual intersection point is going to be 6 comma positive square root of 12? And I should say this is a positive or negative square root of 12. And this row over here is going to be 6 negative square root of 12. This right here is 6 comma 0. That's the center of our circle. So what's the equation of this new circle going to be? What's the equation of this new circle going to be? Well we know its center. Its center is at 6 0. Let me write it down here. Its center is at 6 0. So it's going to be x minus 6 squared plus y minus 0 squared is equal to the radius squared. Now what's the radius? The radius is this height right over here. Or it's just equal to the y value at the intersection. It's the square root of 12. Now this is going to the radius squared. The radius is squared of 12. Squared of 12 squared is 12. So the equation is going to be x minus 6, and see what form they have up here. So they actually multiply everything out. So let's just do that. So this is going to be x squared minus 12x plus 36 plus y squared is equal to 12. And then we could subtract 12 from both sides. Subtract 12 from both sides. And we get x squared minus 12x plus 24 plus y squared is equal to 0. Let's see which of the choices match that. Let me copy it. And then let me paste it up here. Let me paste it up here. And it looks like we have an x squared positive y squared, a negative 12x x and then a positive 24. Negative 12 x positive 24. So it looks like our answer is a. Did I do that right? x squared plus y squared minus 12 x plus 24. Yep, our answer is a.