MAT 6H Rectngulr Equtions of Conics A quick overview of the four conic sections in rectngulr coordintes is presented elow.. Circles Skipped covered in previous lger course.. Prols Definition A prol is the set of ll points P (, ) in the plne such tht d( P, F) d( P, D), where F is fied point in the plne nd D is fied line in the plne. These re respectivel clled the focus nd the directri of the prol. A prol is then the set of ll the points tht re equidistnt to oth the focus nd the directri. is the verte of the prol. This is the point where the prol intersects its is of smmetr (See Fig. 3 on p. 655). Note tht the verte is the point on the prol tht is closest to the directri. Here we let d(, F). If (0,0), these re the four cses: Focus Eqution of Directri Eqution of Prol Ais of Smmetr Description of Prol F (,0) 4 -is Opens right F (,0) 4 -is Opens left F (0, ) 4 -is Opens up F (0, ) 4 -is Opens down
If ( h, k), these re the four cses: Focus Eqution of Directri Eqution of Prol Ais of Smmetr Description of Prol F ( h, k) h ( k) 4 ( h) Prllel to -is Opens right F ( h, k) h ( k) 4 ( h) Prllel to -is Opens left F ( h, k ) k ( h) 4 ( k) Prllel to -is Opens up F ( h, k ) k ( h) 4 ( k) Prllel to -is Opens down Emple Anlze the eqution 4( ) 0. Answer Here we hve 4( ) 0 4 4 0 ( ) 4 4 0 ( ) 4 4 ( ) 4( ) Thus we infer the following: (, ) F (,0) D: This descries prol tht opens up with verte t (, ). Note tht to find the two -intercepts of this prol, it suffices to plug-in 0 in the eqution of the prol. This ields ( ) 4(0 ) 4, which implies tht, or 4,0. So the two -intercepts re the points ( 4,0) nd (0,0). [Grph the prol to confirm ll these nlticl results.] Applictions Proloids of revolution with reflection propert used in the design of stellite dishes, flshlights, hedlights, solr energ devices, telescope mirrors, etc.
3. Ellipses Definition An ellipse is the set of ll points P (, ) in the plne, the sum of whose distnces from two fied points F nd F, clled the foci, is constnt. Here we let d( P, F) d( P, F), where 0. C is the center of the ellipse, nd, re the two vertices of the ellipse. The vertices re the points where the ellipse intersects its mjor is. Note tht C,,, F nd F re ll points on the mjor is. Here we let d( C, ) d( C, ), c d( C, F) d( C, F), nd denote the distnce etween the center of the ellipse nd the two points where the ellipse intersects its minor is (See Figs. 9 & 0 on p. 665). If C (0,0) nd the mjor is is long the -is, we then hve the following: F ( c,0) F ( c,0) (,0) (,0) Eqution of ellipse: where 0 nd, c (See Fig. on p. 666) If C (0,0) nd the mjor is is long the -is, we then hve the following: F (0, c) F (0, c) (0, ) (0, ) Eqution of ellipse: where 0 nd, c (See Fig. 5 on p. 668) If C ( h, k) nd the mjor is is prllel to either the -is or the -is, then replce with h (horizontl shift) nd with k (verticl shift) in the previous equtions for ellipses. This is ll summrized in Tle 3 on p. 670.
Emple Anlze the eqution 4 8 4 4 0. Answer Here we hve 4 8 4 4 0 4 8 4 4 0 4( ) 4 4 0 4( ) 4 ( ) 4 4 0 4( ) ( ) 4 ( ) ( ) Thus we infer tht this is n ellipse with center C (, ) whose mjor is is prllel to the - is (it s the verticl line ) nd whose minor is is prllel to the -is (it s the horizontl line ). We lso hve the following: c 3 Therefore the vertices of the ellipse re the points (,0) nd (, 4), nd its foci re the points F (, 3) nd F (, 3). Note tht since, the two points where the ellipse intersects its minor is re given (0, ) nd (, ). [Check this with the eqution of the ellipse.] [Grph the ellipse to confirm ll these nlticl results.] Applictions Ellipses re used to model plnetr orits nd to design whispering glleries, stdi nd rcetrcks.
4. Hperols Definition A hperol is the set of ll points P (, ) in the plne, the difference of whose distnces from two fied points F nd F, clled the foci, is constnt. Here we let d( P, F) d( P, F), where 0. C, the center of the hperol, is the point where the trnsverse is nd the conjugte is of the hperol meet t right ngle. nd, the two vertices of the hperol, re the points where the hperol intersects its trnsverse is. Note tht C,,, F nd F re ll points on the trnsverse is. Here we let d( C, ) d( C, ) nd c d( C, F) d( C, F). Note tht hperol consists of two seprte rnches (See Figs. 34 & 35 on p. 677) nd hs two olique smptotes (See Fig. 43 on p. 68). If C (0,0) nd the trnsverse is is long the -is, we then hve the following: F ( c,0) F ( c,0) (,0) (,0) Eqution of hperol: where, c (See Fig. 36 on p. 678) Its two olique smptotes re the lines nd. If C (0,0) nd the trnsverse is is long the -is, we then hve the following: F (0, c) F (0, c) (0, ) (0, ) Eqution of hperol: where, c (See Fig. 40 on p. 680) Its two olique smptotes re the lines nd.
If C ( h, k) nd the trnsverse is is prllel to either the -is or the -is, then replce with h (horizontl shift) nd with k (verticl shift) in the previous equtions for hperols. This is ll summrized in Tle 4 on p. 683. Emple Anlze the eqution 4 ( 8 ) 0. Answer Here we hve 4 ( 8 ) 0 4 6 0 4( 4 ) 0 ( ) 4( ) 6 0 ( ) 4( ) 4 0 4( ) ( ) 4 0 4( ) ( ) 4 ( ) ( ) Thus we infer tht this is hperol with center C (,) whose trnsverse is is prllel to the -is (it s the verticl line ) nd whose conjugte is is prllel to the -is (it s the horizontl line ). We lso hve the following: c 4 5 Therefore the vertices of the hperol re the points (,) nd (,3), nd its foci re the points F (, 5) nd F (, 5). Moreover, the hperol hs two olique smptotes: 5 3 or nd. [Grph the hperol to confirm ll these nlticl results.] ( ) nd ( ), Applictions Hperols re used to locte sound nd lightning strikes; hperoloids re used for the design of nucler cooling towers nd mirrors.