PP Section 11.4

# PP Section 11.4 - The Ellipse Definition An ellipse is the...

This preview shows pages 1–8. Sign up to view the full content.

The Ellipse

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Definition An ellipse is the set of points in a plane the sum of whose distances from two fixed points F 1 and F 2 is a constant. These points are called the foci.
To obtain the equation for an ellipse, we place the foci on the x -axis at the points (– c , 0) and ( c , 0) so that the origin is halfway between the foci.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
If P ( x , y ) is a point on the ellipse, the definition says that |PF 1 | + | PF 2 | = 2 a Let the sum of the distances from a point on the ellipse to the foci be 2 a > 0. 2 2 2 2 ( ) ( ) 2 x c y x c y a + + + - + =
2 2 2 2 ( ) ( ) 2 x c y x c y a + + + - + = 2 2 2 2 ( ) 2 ( ) x c y a x c y - + = - + + To simplify algebraic operations, let’s place a radical on each side of the equation. Squaring both sides, we have: 2 2 2 2 2 2 2 2 2 2 4 4 ( ) 2 x cx c y a a x c y x cx c y - + + = - + + + + + +

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 2 2 ( ) a x c y a cx + + = + After simplifying, the last expression becomes: 2 2 2 2 4 2 2 2 ( 2 ) 2 a x cx c y a a cx c x + + + = + + Squaring both sides again to eliminate the radical: After simplifying, the last expression becomes: 2 2 2 2 2 2 2 2 ( ) ( ) a c x a y a a c - + = -
In Geometry we learned that in any triangle the sum of the lengths of two sides is greater that the length of the third side. Consider the triangle determined by the foci and P.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern