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# Common tangent of circle & hyperbola (2 of 5)

## Video transcript

now that we have a visual sense of what this common tangent with a positive slope would look like let's see if we can get some constraints on it especially constraints on its slope and y-intercept so this line this line that I drew in the last video here in pink it would have the form y is equal to MX plus B it's a line where m is the slope and B is the y-intercept where B is the y-intercept now what I let's think about what what constraints that would have to be on M and B if this is tangent if this is tangent to the circle and you might be tempted to break out some calculus and figure out the slope at any point along side a circle but there's an easier way to do it you just have to realize that if a line is tangent to a circle it will only intersect that circle at one point let me show you what I'm talking about so this is the line what I want to do is figure out where this equation and the equation of the circle intersect that's what we'll focus on in this video and then we'll do the same thing for the hyperbola so we have Y is equal to MX plus B is the equation of the line the circle they give us the equation up here x squared plus y squared minus 8x is equal to 0 so the circle is x squared plus y squared minus 8x is equal to 0 so we have what we can do is we can substitute this expression over here we can substitute this in for Y and then we can figure out what does M and B or what are the constraints on M and B so that we only have one solution to the intersection where we only intersect at one point so to do that actually let's substitute for Y squared so if Y is equal to this let's square that we'll get Y squared I'm just squaring the expression for the line y squared is going to be cool M squared x squared plus 2 m bx plus b squared all i did is i squared this expression here and I did that so now we can substitute this whole thing right in here right in here for Y squared and so our the X point the expression for the ex point for our intersection is going to be x squared plus all of this business that's the y squared plus M squared x squared plus 2 m bx plus b squared minus 8x minus 8x is equal to 0 and if we wanted to write this as a quadratic in terms of X this would be so our x squared terms are these 2 terms so this is we could write this as let's see let's write this is M squared plus 1 times x squared right M squared plus 1 times x squared and then our X terms are this one and this one so then we have plus plus 2 m be minus 8 times X and then we just have this B term this B squared the constant term right over here and I'll do that in orange so plus B squared is equal to 0 so if we knew m and B if we knew the equation of this line this would just be a straight-up quadratic you could use the quadratic formula to figure out the x-values where they intersect now what's neat about this is we know that they only have to intersect in one point remember the quadratic formula negative b plus or minus the square root of b squared minus 4ac all of that over 2a and I'll get this be confused with the y-intercept B this is just the B from the quadratic formula that's the quadratic formula over there this will only have one solution will only have one solution if this over here is equal to if this over here is equal to 0 because then you're just put you're just adding and subtracting 0 so you're only going to get one solution so when something when a line is tangent to a circle it can only intersect in one point or another way to think about this will only have one solution if a line intersects any other type of non tangent line a non tangent line would do something like that it would either have two solutions in which case this is a positive value or it won't intersect at all or it won't intersect at all and then this will have no solutions which means that b squared minus 4ac would be a negative number so we know that this is a tangent line this is a tangent line so we only have one solution or b squared minus 4ac is equal to zero so what's B squared minus 4ac over here well this is our B but when we think in terms of the quadratic formula and remember don't get that be confused with the B of the y-intercept I'm just thinking of the quadratic formula here so let's do this let's take this squared so I'm just going to rewrite this expression and set it equal to zero because we know there's only one solution so we have to M be minus eight squared and then you have minus four minus four times a which is M squared plus one times C times C is V squared times C is B squared and so this is going to have to be equal to zero if this is truly a tangent line if this is truly a tangent line so let's see any type of interesting things that we can get out here if we can have if we can express B and as a function of M let that's a good place to start so let's try to do that so let's see if we expand this out this becomes four M Squared let me do that in the same blue so you know what I'm expanding out this part over here becomes four M Squared B squared for M Squared B squared minus two times 8 is 2 times negative eight is negative 16 multiply that times 2 so it's negative 32 MD I'm just squaring this over here plus 64 so that is that term over there expanded - minus 4 times let me write this well I can just expand everything out minus 4 times M Squared B squared minus 4 times 1 times B squared minus 4 B squared is all going to be equal to 0 and then lucky for us some of these terms cancel out 4 MB squared negative 4 MB squared and let's see we could actually divide everything both sides of this equation by 4 and we get we get negative 8 M b-plus we're dividing everything by 4 so plus 16 minus B squared is equal to 0 and now we can solve for B in terms of M using the quadratic formula again so now we would have a constraint or we would essentially know what our y-intercept is going to be in terms of our slope and then we can do that with a hyperbola and then we can essentially say well it's the same line so the y-intercepts have to be the same and then we can solve for the slope so let's do that and you'll see that over the next few videos so let me just write this in a form that we would recognize this is the same thing well let me just let me just actually and let me just multiply this equation right here both sides by negative 1 so then it would become b squared plus 8 m b minus 16 is equal to 0 i just multiplied this by negative 1 and just rearranged the terms now let's solve for B in terms of M so B is going to be equal to negative 8m negative 8m plus or minus the square root plus or minus the square root of this term squared so it's 8 squared M squared minus 4 times a which is just 1 times C which is 16 minus 4 times negative 16 so this is you could view this as plus 4 times 16 plus 4 times plus 4 times 16 all of that all of that over 2a well a here is just 2 all of that over 2 now this is going to be equal to negative 8m plus or minus now this is 64 this is 64 so we can take out you could factor out the 64 from here but when you take the square root of it it's going to be 8 times M Squared 8 times M squared plus 1 right if you took the 8 in you would have to square it so it becomes 64 and 64 times M squared plus 64 which exact which is exactly what you had up there all of that all of that over 2 and then we can simplify this is equal to negative 4 plus or minus four times the square root of M squared plus one so this is this is a possible B given that the line is tangent given that the line is tangent to the circle now let's just think about this a little bit if we if we add four if we add four we're definitely going to have well let's think about it for a second if we look at the line up here the way I draw we want a positive slope we want a positive slope and in order to do that the way I drew it you have to have a positive y-intercept so this is a positive let me just write it this way this is a positive this is a positive be a positive y-intercept so we want this value we want to think about the y-intercept that is positive now M is going to be positive we know from the problem that we're looking for a positive slope so M is positive so negative for this whole term is here is going to be negative so only chance of being positive is so if we add four times this expression right over here and actually if you look at it it will be positive because this is greater than M Squared so this is going to be the square root of that's going to be greater than M so four times this is going to be greater than four M so if we add it's going to be positive so we want to only look at the B is equal to negative four M plus 4 times the square root of M squared plus one I'll leave you there for this video in the next video we're going to do the exact same thing for the hyperbola realizing that the line will only intersect at one point and then since it's the same line we know that their B's have to be the same we're going to in the next video we're going to get a B is equal to some other function of M is equal to some other function of M we're going to get that in the next video and then we could set them equal to each other and solve for our M and then once you solve for an M you also have solved for the B and we'll have our line