00:01
So here we're given a diagram of a circle and it says here that we have a circle that is centered at 5, negative 4.
00:10
It touches the y -axis at point d.
00:14
So that's a point d.
00:15
Since it just touches there, we can assume that that is a tangent.
00:20
And if it's tangent, then that means it has to come out here.
00:27
The radius would have to meet.
00:29
So if i draw the center and i draw a radius over to the d, then that is perpendicular to the y -axis.
00:37
And that's key here because then that means that the 5, i'm going to count over 5 and then down 4 to get that center point.
00:47
So i know what my radius is.
00:50
My radius here is 5.
00:53
The length of the radius here is 5.
00:57
So what's the equation of the circle? well, the equation of the circle would be x minus 5 squared plus y plus 4 squared equals the radius squared, which would be 25.
01:12
5 squared.
01:13
And if you're wondering, you just take your x value and you change it to a negative.
01:19
Take your y value and change it to a negative right there.
01:25
Or you change the negative to a positive right there.
01:28
You just change the signs on both of them.
01:29
You say x, the parentheses and the squared and the x, that's going to be there the whole time.
01:35
And the plus, that's just the equation of the circle.
01:39
So these numbers here will be the center.
01:41
And this number right here is the radius squared.
01:45
So what are these points a and b out here? what's their coordinates? well, i do know that point b here is going to have some x value, but then i know the y value is 0.
02:00
I know the y value is going to be 0.
02:02
So if i know the y value is 0, why don't we take our equation of a circle then and put the 0 in for y.
02:17
Plus 4 squared equals 25.
02:20
And then we'll just solve this.
02:22
And it'll tell us the x value because that's the equation tells me the points on circles.
02:25
That's what the equation does.
02:29
So just rewrite that part.
02:32
0 plus 4 is 4.
02:34
We square it and get 16.
02:37
That's 25.
02:38
Let's go ahead and take that 16 away.
02:41
So x minus 5 squared equals 9.
02:45
We'll take the square root to get rid of that to the power of 2.
02:48
Now when i take a square root, we have to say plus minus 3.
02:54
And when i do that, it'll be like, i'll have to add 5 to get rid of the 5 over there.
03:00
So i'll have x equals positive 3 plus 5, which is 8.
03:05
And x equals negative 3 plus 5, which equals 2.
03:12
So this point here would be 8, 0.
03:18
And this one over here, a, would be 2 comma 0.
03:24
It would be 2 comma 0.
03:26
According to the points a and b, we've got 2, 0 and 8, 0.
03:32
What about the equation for the tangent b? so let's get rid of all that stuff.
03:41
And i look here at the diagram.
03:42
It says that point b is a tangent line.
03:48
And it goes through the x -axis here, point p.
03:52
So what we know is that this is tangent.
03:56
So remember, if i draw a radius to here, then this has to be perpendicular.
04:02
And what i know about perpendicular lines is that when i multiply the slope of two lines that are perpendicular, they have to multiply to give me 1.
04:14
So for instance, if i knew the slope of one line was another line, and they were perpendicular to each other, if i knew the slope of this one was, i don't know, 1 half, then the other one has to be the opposite of that, which is negative.
04:33
And then it has to be reciprocal.
04:36
So you flip it.
04:39
And i know that because 1 half times a negative 2 over 1 will give me negative 1.
04:46
So when i multiply the both of them together, i should get a negative 1.
04:51
One should be positive, one should be negative.
04:53
So i should get a negative number.
04:55
And then when i multiply them both together, i should get a negative 1.
05:01
So we can find the slope of this radius, because i know the point b here is a comma 0.
05:08
I know that there's another point on that line.
05:11
If i know that, then i can find the slope of the other tangent line.
05:17
So basically what i'm saying is, if i could find that this slope is negative 2 over 1, then i would make the opposite reciprocal, and i can get the other slope of that line.
05:26
So i want to find the slope of that tangent line.
05:28
So let's find the slope of the radius...