Geometry: An Interactive Journey to Mastery

Lesson 29: Folding and Conics

Example 1

3RLQWLQJDÀDVKOLJKWGLUHFWO\WRZDUGDZDOOSURGXFHVDFLUFOHRIOLJKWDJDLQVWWKHZDOO

ͼ௘6HHFigure 29.4௘ͽ

D௘ͽ :KDWVKDSHRIOLJKWDSSHDUVRQWKHZDOOLI\RXDQJOHWKHÀDVKOLJKWVOLJKWO\"

ͼ௘6HHFigure 29.5௘ͽ

E௘ͽ :KDWVKDSHRIOLJKWDSSHDUVRQWKHZDOOLI\RXSODFHWKHÀDVKOLJKWYHUWLFDOO\

DJDLQVWWKHZDOO"ͼ௘6HHFigure 29.6௘ͽ

Solution

$ÀDVKOLJKWSURGXFHVDFRQHRIOLJKWDQGHDFKRIWKHVKDSHVRIOLJKWSURGXFHGRQD

wall represents a slice of this cone.

,QD௘ͽWKHFRQLFVHFWLRQSURGXFHGLVDQHOOLSVH

,QE௘ͽWKHFRQLFVHFWLRQSURGXFHGLVͼ௘KDOIRI௘ͽDK\SHUERODͼ௘+RZSUHFLVHO\ZRXOG\RX

QHHGWRDQJOHWKHÀDVKOLJKWLQRUGHUWRVHHDSDUDEROD"௘ͽ

Example 2

Show that the graph of y = x^2 , called a parabola in algebra class, really is a parabola.

ͼ௘6KRZWKDWLWLVDSDUDERODZLWKIRFXV 1 F (^) ̈ ̧©¹§·0, 41 and directrix the horizontal line
y  4.
Solution
/HWͼ௘x, y௘ͽEHDSRLQWRQWKHSDUDERODZLWKIRFXVF (^) ̈ ̧©¹§·0, 41
and directrix y ^14.
Its distance from F is xy^2 ©¹ ̈ ̧§·^142 , and its distance from the
directrix is y^14.
These need to match, giving the equation xy^2  §· ̈ ̧©¹^11442 y.
Figure 29.4
Figure 29.5
Figure 29.6
14 ͼx, yͽ
^14
Figure 29.7