1 00:00:01,060 --> 00:00:08,950 Students in this video, let's discuss reflection of a point about the line now before we start, let 2 00:00:08,950 --> 00:00:11,310 me quickly tell you what this means. 3 00:00:11,680 --> 00:00:13,540 Let me have my access over here. 4 00:00:13,810 --> 00:00:16,450 So this is my Y-axis and this is my X-axis. 5 00:00:16,810 --> 00:00:20,730 Now, say I have a line over here and I have a point over here. 6 00:00:21,040 --> 00:00:29,320 Now, the reflection of this point about this line means I need to find a point which is of the same 7 00:00:29,320 --> 00:00:33,040 length from this line as this point over here. 8 00:00:33,280 --> 00:00:35,340 And it should be on the other side of the line. 9 00:00:35,530 --> 00:00:40,980 That is, I need to find this point over here, which is of the same length. 10 00:00:41,110 --> 00:00:47,190 If this is X, this also should be X and it should be on the other side of this line. 11 00:00:47,410 --> 00:00:51,400 That is what we mean with the reflection of a point about the line. 12 00:00:51,610 --> 00:00:52,080 All right. 13 00:00:52,240 --> 00:00:53,380 Now let's proceed. 14 00:00:53,570 --> 00:00:57,520 We will classify these type of cases into three types. 15 00:00:58,000 --> 00:01:05,410 The first type is finding the reflection of a point about the line Y is equal to X or Y is equal to 16 00:01:05,410 --> 00:01:06,060 minus six. 17 00:01:06,370 --> 00:01:13,440 The second case is finding the reflection of a point about Y X is equal to zero or Y is equal to zero. 18 00:01:13,450 --> 00:01:15,200 So these are also lines, OK? 19 00:01:15,520 --> 00:01:20,480 And the third case is the general case where we find the reflection about any point. 20 00:01:20,740 --> 00:01:22,570 Now, why do I classify it like this? 21 00:01:22,870 --> 00:01:26,670 It's because these two cases can be solved very quickly. 22 00:01:26,700 --> 00:01:26,850 Right. 23 00:01:26,900 --> 00:01:29,410 You just need five seconds to find the answer. 24 00:01:29,590 --> 00:01:30,040 All right. 25 00:01:30,220 --> 00:01:32,970 Let's start with these two cases over here. 26 00:01:33,100 --> 00:01:35,400 I have my X and Y axis. 27 00:01:35,650 --> 00:01:38,070 Now, first, let me draw these four lines. 28 00:01:38,440 --> 00:01:41,830 This is the line Y is equal to X. Now, why is that? 29 00:01:41,830 --> 00:01:46,490 So you will see that when X is equal to one, Y also will be equal to one. 30 00:01:46,690 --> 00:01:50,710 Similarly, when X is equal to Y, you also will be equal to two. 31 00:01:50,710 --> 00:01:51,050 Right? 32 00:01:51,220 --> 00:01:57,310 So that's why this is the nine Y is equal to X and C over here you have no Y intercept. 33 00:01:57,550 --> 00:02:00,840 That's because it passes through the origin already. 34 00:02:01,090 --> 00:02:04,930 Now this is the equation of why is to X Y is equal to X. 35 00:02:05,170 --> 00:02:10,360 Now let's proceed and draw the equation for Y is equal to minus X. 36 00:02:10,630 --> 00:02:14,590 If Y is equal to minus X, the equation will be this one. 37 00:02:15,620 --> 00:02:20,450 Why is that so when X is equal to one, Y has to be minus one, right? 38 00:02:20,570 --> 00:02:25,940 So X is one, why is minus one when X is equal to Y has to be minus two. 39 00:02:26,270 --> 00:02:26,660 Right. 40 00:02:26,870 --> 00:02:30,320 And when X is equal to minus one, Y has to be one. 41 00:02:30,580 --> 00:02:35,210 So that's why this is the equation for Y is equal to minus X. 42 00:02:35,210 --> 00:02:37,310 So we draw this line like this. 43 00:02:37,640 --> 00:02:39,460 All right, now let's proceed. 44 00:02:39,470 --> 00:02:46,670 We have seen visually how we represent Y is equal to X and Y is equal to minus X. 45 00:02:46,880 --> 00:02:50,690 Now can you tell me what would be the nine X is equal to zero. 46 00:02:51,290 --> 00:02:56,480 Yes, the line X is equal to zero would be the Y axis. 47 00:02:56,480 --> 00:02:56,950 Right. 48 00:02:56,960 --> 00:03:01,480 And similarly, the line Y is equal to zero will be the x axis. 49 00:03:01,610 --> 00:03:01,990 Right. 50 00:03:02,660 --> 00:03:03,390 Do you agree. 51 00:03:03,560 --> 00:03:04,550 Yes, that's true. 52 00:03:04,550 --> 00:03:04,850 Right. 53 00:03:05,000 --> 00:03:05,300 Big. 54 00:03:05,330 --> 00:03:10,750 Any point on the x axis, for example, this one let this point before zero. 55 00:03:11,030 --> 00:03:13,000 You can see that Y is equal to zero. 56 00:03:13,010 --> 00:03:13,250 Right. 57 00:03:13,550 --> 00:03:17,000 Or take this point that this V to come over here. 58 00:03:17,000 --> 00:03:18,670 Also, you can see why is equal to zero. 59 00:03:18,860 --> 00:03:22,130 That's why this is the nine Y is equal to zero. 60 00:03:22,340 --> 00:03:23,610 Similarly big. 61 00:03:23,630 --> 00:03:29,450 Any point on the Y axis, for example, this one let it be three comma zero comma three. 62 00:03:29,450 --> 00:03:37,310 Right over here you can see X is zero or take this point, let it be minus zero, comma minus four over 63 00:03:37,310 --> 00:03:37,520 here. 64 00:03:37,520 --> 00:03:39,040 Also you can see X zero. 65 00:03:39,050 --> 00:03:39,400 Right. 66 00:03:39,560 --> 00:03:45,240 Therefore this is the nine X is equal to zero and this is the nine Y is equal to zero. 67 00:03:45,660 --> 00:03:47,990 Alright, now let me clear up the space. 68 00:03:49,640 --> 00:03:55,250 It's always good to visually understand why these methods work so that you will be able to retain and 69 00:03:55,250 --> 00:03:56,180 apply them, right. 70 00:03:56,450 --> 00:03:56,840 All right. 71 00:03:57,020 --> 00:03:57,800 Now we have. 72 00:03:59,140 --> 00:04:01,150 Drawn the four lines over here. 73 00:04:01,540 --> 00:04:08,110 OK, now let's identify a point three comma for now, we are going to find the reflection of this point 74 00:04:08,260 --> 00:04:10,360 about these four lines. 75 00:04:10,630 --> 00:04:11,050 All right. 76 00:04:11,230 --> 00:04:13,550 We start with why is equal to it. 77 00:04:13,580 --> 00:04:18,880 First, I will tell you the method that is the trick, and then we will think about it and understand 78 00:04:18,880 --> 00:04:19,790 why it is true. 79 00:04:20,050 --> 00:04:25,810 Now, if you want to find the reflection of three comma for about why is equal to X, all you need to 80 00:04:25,810 --> 00:04:33,040 do is you need to interchange X as Y and Y as X, that is for comma three will be the reflection. 81 00:04:33,310 --> 00:04:34,350 Now why is that so? 82 00:04:34,630 --> 00:04:41,290 Now if you draw a perpendicular line through three comma four that's perpendicular to Y is equal to 83 00:04:41,290 --> 00:04:47,830 X, it will look like this and you will see that this distance and this distance over here will be equal. 84 00:04:48,040 --> 00:04:52,960 And remember, that is what we mean with reflecting a point about the line. 85 00:04:53,260 --> 00:04:58,730 So the reflection of three comma for about Y is equal to Xs for comma three. 86 00:04:59,080 --> 00:04:59,530 All right. 87 00:04:59,800 --> 00:05:06,940 Now let's proceed and find the reflection of three comma for about Y is equal to minus X. 88 00:05:09,940 --> 00:05:17,140 Now, if you want to find that all you need to do is you need to interchange why ESX and excess weight 89 00:05:17,350 --> 00:05:23,170 and you need to change the science, that is the reflection of three Khama four about why is equal to 90 00:05:23,170 --> 00:05:27,160 minus six is equal to minus four comma minus three. 91 00:05:27,430 --> 00:05:28,750 Now the reason is the same. 92 00:05:28,990 --> 00:05:34,870 If you draw a perpendicular from three comma four to one minus X. 93 00:05:34,870 --> 00:05:41,560 Right, it will be like this and you will see that this point is at the same perpendicular distance 94 00:05:41,560 --> 00:05:45,580 from Y is equal to minus six as this point over here. 95 00:05:46,090 --> 00:05:46,570 All right. 96 00:05:48,380 --> 00:05:49,200 Let's proceed. 97 00:05:49,660 --> 00:05:56,340 Now, if you want to find the reflection of three comma for about X is equal to zero, right? 98 00:05:56,760 --> 00:05:57,990 That's this line over here. 99 00:05:58,370 --> 00:06:00,080 Let's draw a perpendicular. 100 00:06:00,090 --> 00:06:00,890 That is true. 101 00:06:01,160 --> 00:06:02,120 Three, four. 102 00:06:02,240 --> 00:06:04,480 And it's perpendicular to X is equal to zero. 103 00:06:04,790 --> 00:06:05,040 Right. 104 00:06:05,120 --> 00:06:06,080 It will look like this. 105 00:06:06,080 --> 00:06:06,410 Right. 106 00:06:06,860 --> 00:06:14,810 That is the Y value y coordinate remains the same and you get the answer as minus three, comma four. 107 00:06:15,020 --> 00:06:19,880 So you just need to find the minus of the X value and the Y value remains the same. 108 00:06:20,190 --> 00:06:26,060 If you remember it using this diagram, then you will not need to buy it and you will not make a mistake. 109 00:06:26,310 --> 00:06:30,500 So you keep Y the same and X becomes minus X. 110 00:06:30,500 --> 00:06:32,220 And visually, this is the reason, right? 111 00:06:32,480 --> 00:06:38,060 We have drawn a line perpendicular to X is equal to zero through three comma four, and it should be 112 00:06:38,060 --> 00:06:42,470 at the same distance like this distance over here is equal to three. 113 00:06:42,470 --> 00:06:42,870 Right. 114 00:06:43,160 --> 00:06:46,070 So this distance also should be equal to three. 115 00:06:46,190 --> 00:06:51,560 And because it is on the left side over here, it has to be minus three. 116 00:06:51,560 --> 00:06:51,780 Right. 117 00:06:51,800 --> 00:06:56,900 That's why we have minus three over here and we have minus three four, which is the reflection of three 118 00:06:56,900 --> 00:06:59,690 comma for about X is equal to zero. 119 00:07:00,050 --> 00:07:00,490 All right. 120 00:07:00,680 --> 00:07:05,570 Now let's take the reflection of three comma for about Y is equal to zero. 121 00:07:05,900 --> 00:07:08,700 Now, for that, can you tell me what you need to do? 122 00:07:08,960 --> 00:07:14,360 You just need to keep X constant and take the negative value of white. 123 00:07:14,990 --> 00:07:21,380 And visually, this is the reason you draw a line through three comma for that is perpendicular to Y 124 00:07:21,380 --> 00:07:22,340 is equal to zero. 125 00:07:22,610 --> 00:07:24,350 But that line would look like this. 126 00:07:24,350 --> 00:07:24,710 Right. 127 00:07:24,890 --> 00:07:28,300 And then this distance over here is equal to four. 128 00:07:28,610 --> 00:07:34,010 So you need to take the same distance over here and hence you will get minus four over here because 129 00:07:34,010 --> 00:07:36,470 it is in the negative y axis. 130 00:07:36,470 --> 00:07:36,840 Right. 131 00:07:37,070 --> 00:07:39,280 So let's quickly revise what we have learned. 132 00:07:40,100 --> 00:07:46,970 We have seen what the meaning of reflecting a point about a line is and we have discussed for special 133 00:07:46,970 --> 00:07:47,450 cases. 134 00:07:47,780 --> 00:07:53,390 If you reflect a point about why is equal to X, you just need to interchange the value of X and Y. 135 00:07:53,660 --> 00:07:58,520 If you reflect about Y is equal to minus X, you interchange and change the sine. 136 00:07:58,790 --> 00:08:06,500 If you reflect about X is equal to zero, you keep the value of Y same and you change the value of X 137 00:08:06,500 --> 00:08:07,730 to minus X. 138 00:08:07,730 --> 00:08:08,110 Right. 139 00:08:08,270 --> 00:08:14,690 And if you reflect about Y is equal to zero, you keep X same and you change Y two minus. 140 00:08:14,720 --> 00:08:17,040 Right now we are left with the general case. 141 00:08:17,240 --> 00:08:20,900 Now let's take up the general case in the next video.