4.1 Circles The First Conic Section

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4.1 Circles The First Conic Section
The basics, the definition, the formula And Tangent Lines

The Standard Form of a circle comes from the Distance Formula
What is the length of the hypotenuse below? y2 y1 x1 x2

The Circle Definition; The set of all points in a plane a given distance {r = radius} from a given point {(h,k) = center}.

The Circle The Picture (x,y) r (h,k)

The Circle The Formula (x,y) r (h,k) What if the center is the origin? What happens to the formula?

Practice problems Find the center and radius of the following:

What did you see on that last problem?
Completing the square is the norm, not the exception. Get ready for it and if you need to, go back in your notes and look it up (Section 2.7). Most problems in these next 4 sections (circles, ellipses, hyperbolas and parabolas) will require completing the square whenever you are asked to graph. The clue is, if there are any linear terms (uh, what??) then be prepared to complete the square.

How to graph? This is probably the easiest part of it all. Essentially, plot the center then r And then label those 4 points. If r is irrational, then estimate its value and plot in the general correct area (and label with radical notation)

Practice Problem Graph
(um… what do I do if the 2nd degree terms have coefficients???)

Notes Plan to complete the square
If the radius is irrational, use it anyway and just estimate its location. If the 2nd degree terms have coefficients, factor them out, complete the square, then divide by coefficients.

Now we will learn to find the equation of their tangent lines
What do you know about a tangent line? So, if it perpendicular to the radius, what do you know about the slopes.

What does a tangent line look like?
(x,y) r (h,k)

Does the radius matter? In any circle, does the radius matter when it comes to the equation of the line tangent? 1. Find the equation of the line tangent to the circle at (2,2) No. It only gives you a location for the tangent point. It doesn’t have any effect on the equation itself.

So, what is the rule? Find the standard form of the circle.
Identify the center. Find the slope of the radius from the center to the given point. Use the slope perpendicular (negative reciprocal) with the point ON THE CIRCLE to find the equation of the line tangent.

More examples Find the equation of the line tangent to the circle at (2,7). Find the equation of the line tangent to the circle at the point in the 4th quadrant where x = 4.