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Current time:0:00Total duration:8:55

Common tangent of circle & hyperbola (3 of 5)

Video transcript

in this video we're going to do with the hyperbola the exact same thing we did with the circle we're going to find constraints on the y-intercept for the tangent line in terms of M but this time we're going to use a hyperbola and then we can set them equal to each other and solve for the M so let's remind ourselves what the equation of the hyperbola is they give it to us right over there it's x squared over 9 minus y squared over 4 is equal to 1 let me write this over here so it is x squared over 9 minus y squared over 4 is equal to is equal I'll write the Y squared minus y squared over 4 is equal to 1 but now we can substitute the Y squared with the y is equal to MX plus B that we got it from the last video and we actually figure out what Y squared was equal to Y squared was we equal to all of this business up here because it's the same line remember we're this is the whole point of it we're trying to find two constraints on that same line so we can rewrite this same thing and actually one thing I want to do is I'm going to multiply both sides of this equation by 36 the common multiple of 9 and 4 so that I can get rid of these fractions and so this is going to become this is going to become 36 let me write it this way so 36 divided by 9 is 4 so it's 4x squared minus 36 divided by 4 is 9 and I would put a Y squared here but we know that Y squared is the exact same thing as this thing over here so Y squared is the same thing as M Squared x squared plus 2 MB X plus B squared and then this is going to be equal to this is going to be equal to remember we multiply both sides of this equation by 36 so this is going to be equal to 30 equal to 36 and let me simplify this we're going to do the exact same thing we're going to be no we know that we know that M and we know that the M and B have to be such that or the line has to have it as a slope and y-intercept we're only intersect with the hyperbola at one point it'll only have one solution to this quadratic in terms of X but let's simplify it first before we worry about that so this is equal to 4x squared minus 9 m squared x squared minus 18 minus 18 M B X minus 9 minus 9 B squared just multiply it there and let me subtract this 36 from both sides so then we have minus 36 is equal to 0 so this is a quadratic in terms of X but let me combine let me combine the various degree term so this these are the x squared terms these are the x squared terms right over here so this is the same thing as 4 minus 9 m squared times x squared and then our only X term is this right over here that's our only X term so this is minus 18 m BX and then our constant terms our constant terms are right over here so this over here is minus 9 minus 9 B squared minus 9 B squared let me write it this way let me write it as minus 9 minus 9 let me write it as let me write it as well I'll just write it minus 9 B squared - 36 I'll just write it like that and of course that is going to be equal to 0 and remember quadratic formula if we wanted to solve for the X's we'd have the quadratic formula but we only want to have one solution so the discriminant part of the quadratic formula is going to equal 0 the b squared minus 4ac is going to be equal to 0 this is exactly what we did in the last video so let's take the b squared minus 4ac and set that equal to 0 and then we'll have our M our constraints on M and B so B squared is and remember don't get the B squared in the quadratic formula confused with the B in the y intercept but this term squared is so this is going to be it's going to be 18 squared M Squared B squared right negative 18 squared is just positive 18 minus 4 times a a is 4 minus 9 M squared times C times C so C is so I can rewrite C as negative negative 9 times negative 9 times what is this negative 9 times B squared B squared plus 4 did I do that right negative 9 times B squared is negative 9 B squared negative 9 times 4 is negative 36 I want to make sure I don't make any careless mistakes and so this becomes so if we just take the negative 9 and the negative 4 they become a positive 36 they become a positive 36 and then we can actually let's see just to simplify things so we don't have to do too much fancy math this 18 squared let's remember let's remember 18 squared is the same thing is well 18 squared is going to be divisible well I won't worry too much about that just yet I just want to make sure that actually let me let me write it this way so 18 squared is 2 2 times 9 times 2 times 9 or another way to think about it it is 4 times 9 times 9 that's the same thing as 18 squared 4 times 9 times 9 now we can divide both of these terms we can and this whole thing remember we want this whole thing to be equal to 0 the discriminant has to be equal to 0 so we can divide both sides of this equation we can divide both sides of this equation by 36 which is the same thing as dividing by 4 times 9 so this term right over here we can get rid of 1/4 and 1/9 and we're going to get we're going to get a 9 9m squared B squared over here and then we divided by 36 so these all go away so it's going to be 9m squared B squared plus this thing times this thing Plus this thing times this thing so let's see what that is so we have a 4 times a B squared so we have plus 4 B squared let me do this in a different color for do it in blue so plus 4b squared and then if 4 times 4 so plus 16 then you have negative 9m squared times B squared so it's negative 9m squared B squared and then you have negative 9m squared times 4 so negative 36 negative 36 M Squared and that's going to be equal to 0 lucky for us that and that cancel out and then we are left with we are left with something and actually what we're left with everything is divisible by 4 so let's divide what we're left with by 4 so then we're left with B squared V squared that's that term and then minus 9m squared minus minus 9m squared that's that term over there just divide it by 4 and then plus 4 plus 4 is equal to 0 and once again we could use we could well actually here we don't have to do anything we'd have to use quadratic formula we can just solve for B we could add we could subtract this from both sides and so we'll get V squared V squared is equal to the square root the square root of 9 m squared square root of 9 m squared minus square root of 9 M squared minus 4 sorry that's not B is equal to just write this I don't want to skip steps here V is equal to this B squared is equal to 9 M squared minus 4 they're painting the office right now so maybe it's making me a little bit making my brain not work properly and then B is equal to the square root of 9 m squared minus 4 did I do that right let's see I got the 4 it looks right and so we're left with the situation where the B if we're saying that the line is tangent to the hyperbola has to be this and if the line is tangent to the circle B has to be equal to this this business over here so let me copy it and then let me paste it let me paste it just like that and so we now have two equations with two unknowns we can set these equal to each other and solve for M and this will give us the the M or the slope of that tangent line and then we can go ahead and solve for be I'll do that in the next video