If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Intersection of circle & hyperbola

## Video transcript

the circle x squared plus y squared minus 8x is equal to zero and the hyperbola x squared over nine minus y squared over four is equal to one intersect at the points a and B in problem 46 they tell us equation or they want us to find equation of the circle with a B as its diameter so let's think about let's visualize the circle in 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 plus 16 this over here becomes X minus 4 squared X minus 4 squared plus y squared is equal to 16 and so the circle is going to look something like this that's my x axis that's my Y axis its center is at 4 0 so 1 2 3 4 4 0 and its radius is 4 its radius squared is 16 so it's radius is 4 so let me go 1 2 3 4 1 2 3 4 let me go up 4 1 2 3 4 so all of these points that point that point that point and at that point are all going to be on the circle let me draw that a little bit neater that point that point and at that point are all going to be on our circle so my circle is going to look something 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 minus y so x squared over 9 minus y squared of 4 is going to equal 1 this hyperbola is going to open to the left on 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 get if you you would get let me write it this way negative y squared over 4 is equal to negative x squared over nine plus one 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 it's going to stop mattering so much and it's so as X approaches infinity as X approaches infinity this is going to approach the square root of 4 over 9x squared this constants 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 the slopes let's see 2/3 if you go right if you run 3 you rise 2 so it's going to its 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 & 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 X you just said 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 is 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 a B as its diameter so a B with a B with a B as the diameter I could have drawn that a little bit straighter so the equation of that circle right over there so we essentially just need to figure out where does this hyperbola intersect this circle now if we can the easiest way to do this is to what we have two constraints here let's solve for y squared for the circle and we can substitute that in for y squared right here and see what x values they 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 this the x value comma zero will be the center and then we'll have our equation so let's do that so this up here we have let me 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 negative x squared from or subtract x squared from both sides at X minus x squared I really just moved 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 hyperbolas equation is x squared over 9 minus y squared over 4 is equal to 1 instead of writing 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 in 8x minus x squared and let's see if we can solve 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 set tells us that x squared x squared over 9 minus 8 x over 4 so minus 2 x 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 that's 4 times 9 so 36 divided by 9 is 4 so this is 4x squared minus 36 times 2 is 72 72 X plus 36 divided by 4 is 9 plus 9 x squared is equal to 36 these two terms right here we can add them we get 13 x squared minus 72 X 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 you just put the 36 out back and then all of that all of that over 2 times a so all of that over all of that over 26 so the hard part is really to simplify this but looks like something we have some interesting stuff going on so let me rewrite this part over here so 72 let me write it over here 72 times 72 is the same thing as this 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 for 4 times 36 times 13 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 so we don't waste this real estate over here now we can factor out a 4 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 used the both 2s and a 36 so we have a 36 left we used this 4 times 36 so we have plus 13 so this becomes a square root of 4 times 4 times 36 is 144 36 plus 13 is 49 so it's we're lucky that it actually worked out to two 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 over here this business over here simplifies to 72 plus or minus 84 this whole thing over here simplified to 84 all of that over 26 now if we subtract 84 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 let's only consider let's only consider the plus 84 and see and see what we get so the first thing to do actually let me just divide the numerator in 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 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 30 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 the center is 6 comma 0 but the actual intersection point is going to be 6 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 4 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 six zero its center is at six here let me write it down here its center is at six zero so it's going to be X minus six squared plus y minus zero 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 be the radius squared the radius is square root of 12 square root of 12 squared is 12 so the equation is going to be X minus 6 let's 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 zero and let's see which of the choices 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 and then a positive 24 negative 12x positive 24 so it looks like our answer is 8 I do that right x squared plus y squared minus 12x plus 24 yep our answer is a