The tangent line to a circle may be defined as the line that intersects the circle in a single p

The tangent line to a circle may be defined as the line that...

Key Concepts

Circle

A circle is a fundamental geometric shape defined as the set of all points in a plane that are equidistant from a fixed center. It exhibits important properties related to symmetry, radius, diameter, and circumference, which serve as a basis for understanding more complex geometric principles.

Tangent Line

A tangent line is a straight line that touches a curve at exactly one point without cutting through it. In the context of a circle, the tangent line represents the line that makes contact with the circle at a single, unique point, highlighting its role in approximating the curve locally.

Point of Tangency

The point of tangency is the specific point at which a tangent line touches a circle. This concept is critical in geometry as it illustrates the unique relationship between the tangent line and the circle, notably, the perpendicular relationship between the radius of the circle at that point and the tangent line.

Transcript

00:01 So in this problem, we're giving a little background about a tangent line.

00:05 So the red line is a tangent line.

00:07 It touches the circle only at one point right there.

00:11 And that point is called the point of tangency when that happens.

00:15 So if the equation of the circle is x squared plus y squared equals r squared, and the equation of the tangent line is y equals mx plus b.

00:27 So we're asked first to show that r squared.

00:32 Plus 1 times 1 plus m squared let me clean this up just a little bit here equals b squared okay and we're giving a little hint here right so if i take this equation for y and i plug it in there then i get the equation that's in the hint don't i i get x squared plus mx plus b squared is equal to r squared.

01:12 And this equation at that point of ten content, a point of tangency has only one solution for x.

01:21 So let's figure out what that solution for x is.

01:25 So i have x squared, square everything out in here plus m squared x squared plus two mxb plus b squared and bring the r from over from the other side minus r squared is now equal to zero.

01:46 So now i'm in standard form, aren't i? where i can think of this as 1 plus m squared times x squared plus 2 mb times x plus b squared minus r squared.

02:09 Alright? i can think of it like that.

02:13 So that this is a, this is b, and this is b, and this this is c in the quadratic equation.

02:21 Remember the quadratic equation says that x equals minus b plus or minus the square root, b squared minus 4a, c all over 2a.

02:31 All right.

02:32 So let's plug in everything we know now.

02:35 So this is minus 2mb plus or minus the square root of b squared.

02:48 So that's 4m squared, b squared, minus 4 times a, 1 plus m squared, times c, b squared minus r squared, all over 2 times a, 1 plus m squared.

03:15 Okay, so for this to have only one solution, then everything under the radical, has to equal 0.

03:32 Okay, then 4m squared, b squared, minus 4 times 1 plus m squared times b squared minus r squared equals 0 for that to happen.

03:48 Okay, so let's multiply this out now.

03:53 So i got minus 4 times, let's see, b squared minus r squared plus m squared b squared minus m squared r squared that equals zero so what can i see now this four m squared b squared cancels out with that one doesn't it because it's minus four times it over here all right so then i can also divide everything by minus four what's left here right divide both sides by minus four and so what do i have.

04:33 I have b squared minus r squared minus m squared r squared is zero.

04:44 Well, take all of this to the other side, and what do we get? we get b squared equals r squared plus m squared r squared, which is r squared times 1 plus m squared, which is exactly what we were trying to show...

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