00:01
Okay for this problem we're given that y equals 4x plus 6 and we know that there's a circle with a center of 2, negative 3 and that's the center and we know that somewhere at the center it intersects, it is tangent to the line y equals 4x plus 6 and it's asking us to find the equation of the circle so the equation of a circle.
00:33
Okay so in order to do that well what do we need to do? well we can draw a rough sketch to help us out so 4x plus 6 looks something like this, there's 6 on the y -intercept and then we're going to go up 4 and across 1, down 4, across 1.
00:55
So it should look something like this y plus 4x plus 6, it's just a rough sketch and we know 2, negative 3 is somewhere around here and if we draw a circle around this it would look something like that and intersects somewhere around here so the radius would be this right the center to there is the radius and the center is 2, negative 3.
01:23
So if we want to find the equation circle we're going to use the y equals x minus h squared plus y minus k squared equals r squared.
01:36
Okay so hk represents our center which we already know we just need to solve for r.
01:44
So how can we solve for r? well the equation of this line was y plus 4x plus 6 and we have a point and with a line we can use the distance formula from a point to a line so the formula for it is d equals x1 plus by1 plus c divided by square root of a squared plus b squared by1 plus c divided by a squared plus b squared.
02:25
So x1 y1 is the same as 2, negative 3 that's our point.
02:31
Now we need to rewrite this line into the that form of ax1 plus by1 plus c.
02:36
So y equals 4x plus 6 we can move everything to one side so subtract 4x and subtract 6 and therefore we got ax1 plus by, let me switch that around, we get by1 plus ax1 plus c is zero.
02:58
So b must be 1, a must be negative 4, and c is negative 6...