 So next we are going to talk about the equation of the chord connecting two points having eccentric angles theta and phi. Now eccentric angle is basically nothing but the parameter for that particular point. So let's say we have the circle and we have a chord which connects two points having eccentric angles theta that means in other words this point is A cos theta comma B sin theta sorry A cos theta comma A sin theta and the other one is A cos phi comma A sin phi. Now this result is basically very important that we are going to derive now that's the equation of the chord is going to be x times cos theta plus phi by 2 plus y sin theta plus phi by 2 is equal to A cos theta minus phi by 2. So this is the desired equation and how do we prove this? Very simple you just have to follow the equation of a line in two point form. So y minus y1 let's say y minus A sin theta is equal to slope will be A sin theta minus sin phi by A cos theta minus cos phi times x minus A cos theta. So A and A gets cancelled off and sin theta minus sin phi is going to be 2 cos theta plus phi by 2 sin theta minus phi by 2 divided by this is going to be 2 sin theta plus phi by 2 into sin phi minus theta by 2 or minus sin theta minus phi by 2. So let's cancel off the terms which are not needed okay and let's expand this. So for that I will cross multiply first. So it becomes minus y sin theta plus phi by 2 plus A sin theta sin theta plus phi by 2 and on this side we will end up getting x cos theta plus phi by 2 minus A cos theta cos theta plus phi by 2 correct. Now let's bring this term to this side and this term to this side. So we will end up getting x cos theta plus phi by 2 plus y sin theta plus phi by 2 plus A times cos theta cos theta plus phi by 2 plus sin theta sin theta plus phi by 2. Now this is clearly the right hand side is clearly the formula of cos A cos B plus sin A sin B which is going to be which is going to be A cos theta minus theta plus phi by 2 which is A theta minus phi by 2. So this becomes the equation of a chord connecting two eccentric angles theta and phi and you can all see over here that if your phi approaches theta that means the two points come closer and closer to each other you would realize the equation starts becoming this cos theta plus theta by 2 y sin y sin theta plus theta by 2 is equal to A cos theta minus theta by 2 which is nothing but A cos theta plus y sin theta equal to A that's nothing but equation of tangent tangent at theta. So the limiting case of this chord will become a tangent to the circle at that point theta. Now guys we are going to solve some problems we have discussed enough theory so far and we are good to go with the problems right now we will start with this question prove that the line Lx plus My plus n equal to 0 touches the circle x minus A whole square y minus B whole square equal to r square if La plus mb plus n the whole square is equal to r square L square plus m square very simple question to start with please type done if you are done with this question on the type in the chat on the chat box so that I can start this discussion of this question done very simple if it touches you can say the distance of the center from this particular line is going to be r okay so this distance from a point A comma B from this line is mod La plus mb plus n by under root of L square plus m square and this distance should be equal to r correct so let's square both the sides square both the sides you directly get La plus mb plus n whole square is equal to r square L square plus m square okay hence proved any question with respect to this next is a line x minus 2 cos theta plus y minus 2 sin theta equal to 1 touches a circle for all values of theta for all values of theta okay find the equation of the circle find the equation of the circle basically the question says irrespective of whatever is your theta this line will always touch a circle find the equation of that circle please type done if you are done with this so that I can start the discussion if you are done please type done in the chat box okay let's discuss this so guys if this two were not there then you can say this is going to be a tangent to a circle whose this is tangent to a circle whose center is at origin and radius is going to be 1 correct so it's going to be tangent to x square plus y square is equal to 1 correct now we're replacing x with replacing x with x minus 2 okay and y with y minus 2 what what is what does it mean when you're replacing x with x minus 2 and y with y minus 2 that means your origin is going to minus 2 comma minus 2 right so if your origin is going to minus 2 comma minus 2 means the circle is now coming at 2 comma 2 center is now coming at 2 comma 2 like this isn't it earlier the origin was here earlier the origin was at the center now when you are replacing your x with x minus 2 and y with y minus 2 it implies that you are shifting your origin to minus 2 comma minus 2 that means your center will automatically be shifted to 2 comma 2 correct so I can clearly say that x minus 2 cos theta plus y minus 2 sin theta equal to 1 will be tangent to x minus 2 square y minus 2 square equal to 1 so this becomes my answer okay next question find the equation of tangents to the circle x square plus y square is equal to 16 drawn from the point drawn from the point 1 comma 4 so this is slightly differently framed question so from an external point you are drawing two tangents they're asking the equation of these two tangents okay so this is 1 comma 4 and this equation is your x square plus y square is equal to 16 I know I haven't yet done done with you the equation of pair of tangents but even without that concept you can solve this problem please feel free to type in your answer in the chat box whatever you have got all right so how to do this problem we have already seen that y equal to mx plus minus a under root 1 plus m square is the slope form of the equation of a tangent okay so let us assume that this is a tangent that I get okay to this circle and let me substitute a as 4 because the radius of the circle is 4 units correct now I'm going to take mx on the other side and square it this is what I'll be getting now be informed that this tangent is going to be satisfied by 1 comma 4 so instead of y I'll put 4 instead of x I'll put 1 so this is what I end up getting that is 16 plus m square minus 8 m is equal to 16 plus m square 16 plus 16 m square which means 15 m square is equal to minus 8 m this implies two things one m can be zero and other m can be minus 8 by 15 so the tangent to a circle passing through 1 comma 4 can have two tangents as it is very much evident from this graph so we'll have two lines two tangents tangent t1 and tangent t2 okay and what would be the equation of such tangents be very simple use the slope point form of the equation of a tangent so t1 would be let's say y minus 4 is equal to 0 times x minus 1 that means y is equal to 4 and t2 I'm assuming the other tangent to be named as t2 so y minus 4 is equal to minus 8 by 15 x minus 1 that makes 15 y minus 60 is minus 8x plus 8 that is 15 y plus 8x minus 68 or is equal to 68 is the other equation so these are the two tangents that can be drawn from the point 1 comma 4 to this given circle okay