 So let's move on to the concept of diameter of a parabola Diameter of a parabola So how do you define a diameter of a parabola? So basically diameter is defined as the locus of the middle points Or the midpoints of a system of parallel chords Which is called the diameter So it's the locus of the midpoints of a system of parallel chords Which we call as the diameter So let us take a situation Let's say we have some set of parallel chords And you connect their midpoints And you connect their midpoints So you get a line like this That line will be called the diameter of this parabola Let's say all these lines are having a slope of m So m is the slope of these lines Then prove that Prove that the equation of the diameter is The equation of the diameter is y is equal to 2a by m Where m is the slope of the parallel chords Which the diameter is bisecting Please do that Please do this question Please type done if you are able to solve this Remember you are finding the locus of the midpoints of these chords Having a slope of m So h, k is the midpoint of the chord Rohan, Lalitha, Atmesh Any idea? Okay, very simple Just proceed from the concept of the equation of the chord whose midpoint is known We know that the equation of the chord whose midpoint is known is t equal to s1 So this is your t equal to s1 So s1 will be k square minus 4ah Now according to the given question This line has a slope of m So what is the slope of this line If somebody asks you what is the slope of this line So you will say let me write the line as y equal to mx plus c So you will write it something like this So y is equal to 2a by k Plus this entire thing Which is actually k square minus 2ah by k Now this is going to be the slope This is going to be the slope So compare that with m So we will get m as 2a by k And now when we generalize it we write m as equal to 2a by y Which means y is equal to 2a by m Would be the equation of the diameter Would be the equation of the diameter This would be the equation of the diameter Guys why I am asking you to derive this Is because indirectly it is a locus question Correct So you should be very very well versed How to deal with locus Is it clear please type CLR if it is clear So basically as you can see here That the line is parallel to the axis of the parabola Yes or no So y is equal to 2am is basically Parallel to the axis of the parabola Now there are not many questions based on this So we will directly move on to the concept of Pole and polar So we will move on to the concept of pole and polar Again the concept remains the same As what we had discussed in case of a circle So I will repeat it once again quickly So let's say there is a point X1 Y1 which can be inside Or outside the parabola Let's say this is point X1 Y1 Through this point if you draw infinitely Many quads let's say I draw these two quads And at the ends of these two quads Let's say you draw tangents At the ends of these two quads Let's say you draw tangents Then you would realize that The end points of these all tangents Will start meeting on a line They will start meeting on a line like this And this point is what we call as the pole And this line is what we call as the polar Okay Remember your pole can be outside the parabola as well Let's say you have a situation like this Where the point is outside Let's say this is the point X1 Y1 Now from this point if I draw multiple tangents Multiple quads I am sorry not tangents Multiple quads Let me take this point over here Yeah Let's say this chord you have drawn Then you have drawn another chord like this Okay And at the ends of these quads If you draw tangents If you start drawing tangents Let's say this will cut further somewhere Okay And just roughly drawing the figure And when you connect all these points You would realize that they will lie on the straight line They will lie on the straight line Let me show that with green Okay So this green line would be called as your polar For this pole Okay So now what is the equation of this polar Equation of the polar is given again as T equal to 0 So the same equation T equal to 0 Is now being used for the third time The first time it was used when X1 Y1 was on the parabola And it represented the equation of the tangent Right So remember T equal to 0 represent three things If X1 Y1 is on the parabola It represents the tangent It becomes a tangent When X1 Y1 is inside the parabola It becomes polar And when X1 Y1 is outside the parabola It becomes polar Or it can become chord of contact Or it can become chord of contact Remember chord of contact will coincide with the polar So if I make a chord of contact over here Okay Then this chord of contact that is the white line Will coincide with the polar So chord of contact and polar will be having the same equation Is that fine? So guys we will take some questions But before that we will take two quick properties Which are very obvious ones First being Tangent is the polar of its point of contact Is the polar of its point of contact That means if your pole is X1 Y1 That means if your pole is X1 Y1 And you are drawing a tangent at X1 Y1 Then the tangent itself will behave as the polar Okay So if the point X1 Y1 lies on the circle On the parabola Then the tangent itself This will act as your polar for this pole Next important property is Polar of the focus is the directrix Polar of the focus is the directrix Can you prove it quickly? Polar of the focus is the directrix So if you take the pole as A comma 0 Then polar will be X plus A equal to 0 Can you prove this? Done If you are done piece type done on your screen On your chat box It should take you not more than 5 seconds to solve this problem Done? Yeah, it's very simple All you have to use is T equal to 0 That is Y Y1 is equal to 2A X plus X1 Remember this is your X1 This is your Y1 So it will be Y into 0 Is equal to 2A X plus A That clearly implies X plus A equal to 0 Okay Let's take a question on this And then we will take a break Small break So the question based on polar and polar is this Again a locus question Show that the locus of the poles Of normal chords Of Y square is equal to 4A X X plus 2A Times Y square plus 4A cube equal to 0 Show that the locus of the poles of the normal chords Of Y square is equal to 4A X Is given by X plus 2A times Y square plus 4A cube equal to 0 So basically you have to find the locus of the poles Which are generating normal chords So these are polar So if polar is a normal chord Then what is the locus of the pole? That's what the question asks If normal chord is your polar Then what is the locus of pole? It's a very very simple question Please attempt this If done please write done on the chat box Okay So let's discuss this very simple again Let's say the pole is H comma K Okay Now if pole is H comma K The polar will be what? So let the pole be H comma K The polar will be what? Polar will be T equal to 0 Correct So I can say YK Is equal to 2A X plus H YK is equal to 2A X plus H Okay Now try to compare this with the equation of a normal We all know the equation of the normal is Y plus TX is equal to 280 Plus 80 cube Now according to the question these two things are the same The polar is the normal chord Or the normal chord normal means the same thing Correct So their coefficient should be proportional to each other Right So let me write this equation as Y is equal to 2A X by K Plus 2A H by K Okay So let's compare the coefficient now So if I compare the coefficients I can clearly see that T will be equal to 2A by K And 280 plus 80 cube will be equal to 2A H by K Correct Okay So all you need to do is substitute T over here Which implies 2A into 2A by K Plus 80 cube is equal to 2A H by K So let's simplify it 4A square by K And here I will have 8A to the power 4 by K cube Equal to 2A H by K Or you can drop a factor of 2A from everywhere So you can drop a factor of 2A from everywhere So 2A by K Plus 4A square Sorry 4A cube By K cube 2A H by K So drop a K also from everywhere down Just one minute guys Oh one small mistake Minus T was that Yeah minus sign will come over here Minus sign will come over here as well I'm so sorry I did a mistake in the sign That's why I was wondering it's not matching with the answer Okay yeah Now if you simplify this You are going to get 4A square Plus 2A Plus H times K square Equal to 0 4A cube by the way Yeah That means your equation can be written as This 2A is gone right So this 2A will not be there Sorry yeah So it can be written as X plus 2A times Y square plus 4A cube Equal to 0 Is that fine guys