ELLIPSE A circle under stress!

THE ELLIPSE Though not so simple as the circle, the ellipse is nevertheless the curve most often "seen" in everyday life. The reason is that every circle, viewed obliquely, appears elliptical.

THE ELLIPSE Any cylinder sliced on an angle will reveal an ellipse in cross-section (as seen in the Tycho Brahe Planetarium in Copenhagen).

THE ELLIPSE Tilt a glass of water and the surface of the liquid acquires an elliptical outline. Salami is often cut obliquely to obtain elliptical slices which are larger.

THE ELLIPSE

HOW IS THE ELLIPSE LIKE A CIRCLE? Hmmm…..

TAKE A MINUTE TO WATCH THIS VIDEO How to draw an ellipse… And here is another one if you want to look at it later.

Play around with a virtual ellipse or build your own with thumbtacks and string.

Think about the video and the demonstration about how an ellipse was created. The combined length of the green segment and the red segment remains constant. You should have a good idea of what that constant may be?

THE ELLIPSE The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

THE ELLIPSE Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eaves-dropping on the private conversations of other House members located near the other focal point.

The combined length of the segments is equal to the length of the MAJOR AXIS. Major axis means the BIG one. Minor axis means the small one. An ellipse will always have one axis bigger than another. Otherwise it is a circle…so a circle is just a special ellipse.

CONSIDER THE FOLLOWING The two FIXED POINTS are called the FOCI The foci are ALWAYS on the MAJOR axis for any ellipse.

USE P.T. TO DETERMINE THE COORDINATES OF THE FOCI The dashed red line is the same length as the semi-major axis---in this case….a

THE MAJOR AXIS IS NOW THE VERTICAL AXIS---this is allowed  Just like before, the dashed red line is the same length as the semi-major axis---in this case…. b

ANSWER THE FOLLOWING QUESTIONS BASED ON THE DIAGRAM GIVEN What is the length of the major axis? What is the length of the semi-minor axis? What are the coordinates of positive focus? What is the combined distance from P to both foci?

ANSWER THE FOLLOWING QUESTIONS Consider the diagram at the left, what are the coordinates of the foci? If the focus of an ellipse is (0,8) and the length of the minor axis is 12, then what is the length of the semi-major axis?

SO NOW WHAT DOES THE EQUATION OF AN ELLIPSE LOOK LIKE??? Recall that a circle is really a special ellipse Consider as an example….. Divide both sides of the equation by 25 and see what happens

Does this look like ANYTHING you may have seen last year??? So now we have: Does this look like ANYTHING you may have seen last year???

HOPEFULLY YOU SAID THAT IT LOOKS DECIDEDLY LIKE …THE SYMMETRIC FORM OF THE EQUATION OF A LINE, WHERE YOU WILL RECALL THAT …. What is under the x is your x intercept What is under the y is your y intercept

GOING BACK TO OUR CIRCLE What is under the x2 determines your x intercepts…we need to square root 25 to get the x intercept. What is under the y2 determines your y intercepts…we need to square root 25 to get the y intercept

NOW THINKING OF AN ELLIPSE where the x- and y- intercepts are DIFFERENT from each other…. Clearly, we will have different values under the x2 and y2 terms in the equation to reflect the different x- and y-intercepts What is under the x2 determines your x intercepts…we need to square root 4 to get the x intercepts What is under the y2 determines your y intercepts…we need to square root 9 to get the y intercepts

EQUATION OF AN ELLIPSE IN STANDARD FORM Where a2 determines the length of the horizontal axis Where b2 determines the length of the vertical axis If the value of a is larger then the horizontal axis is the major axis and the length of the major axis is 2a If the value of b is larger then the vertical axis is the major axis and the length of the major axis is 2b

JUST A QUICKIE The length of the major axis is 20 and the length of the minor axis is 16 and the focus is on the horizontal axis

PUTTING IT ALL TOGETHER What are the foci? solution So the foci are and

Page 332 #3 and #4 Page 333 #5, #6 and #7 HOMEWORK Please be sure to check your answers on the class website before next class. That way you can come prepared with your concerns and questions.