Study Sheet ( )

Transcription

1 Key Terms prol circle Ellipse hyperol directrix focus focl length xis of symmetry vertex Study Sheet ( ) Conic Section A conic section is section of cone. The ellipse, prol, nd hyperol, long with few other mthemticl shpes, cn ech e seen to e section of cone. Prol A prol is the set of ll points ( y) x, tht re the sme distnce from fixed line (clled the directrix) nd fixed point (focus) not on the directrix. A prol is not simply one rnch of hyperol. Indeed, the rnches of hyperol pproch liner symptotes, while prol does not do so. The vertex is midpoint etween the focus nd directrix of the prol. The xis of the prol is line pssing through the focus nd vertex. The Eqution of Prol Given focus ( h k + p), nd directrix y = k p the eqution of prol is Where = nd is the vertex. y = ( x h) + k,

2 Key Terms Ellipse Mjor xis Focus (plurl: foci) Minor xis Vertex Ellipse An ellipse is the set of points in plne the sum o f whose distnces from two fixed points F 1 nd F is constnt. These two fixed points re clled the foci. The Eqution of n Elipse with Center (0,0) The stndrd eqution of n ellipse centered t the origin is given y where nd re some positive constnts. Vertex The x-intercepts re found y setting y=0. The corresponding points (,0) nd (-,0) re clled the vertices of the ellipse. Mjor Axis The line segment joining the vertices, (,0) nd (-,0), is clled the mjor xis. Center The midpoint of the mjor xis is clled the center of the ellipse. Minor Axis The minor xis is the line segment perpendiculr to the mjor xis which lso goes through the center nd touches the ellipse t two points., The Eqution of n Elipse with Center (h,k) The eqution of n ellipse centered t the point (h,k) is given y where nd re some positive constnts.,

3 Key Terms Hyperol Trnsverse xis Fundmentl rectngle Conjugte xis Hyperol A hyperol is the set of ll points in plne the difference of whose distnces from two fixed points (the foci) is constnt. The Eqution of Hyperol Centered t (0,0) Opening Left nd Right x y The hyperol, hs foci (±c,0),where =, vertices (±,0), nd symptotes y ± x The Eqution of Hyperol Centered t (0,0) Opening Up nd Down y x The hyperol Hs foci (o,±c),where Fundmentl Rectngle =, vertices (0,±), nd symptotes y ± x The rectngle with four points (,), (,-) (-,) nd (--) is clled the fundmentl rectngle. Conjugte Axis The line segment joining the endpoints, (0,) nd (0,-), is clled the conjugte xis. The Eqution of Hyperol Centered t (h,k) Opening Left nd Right The hyperol hs foci (h±c,k),where ( x h) ( y k) =, vertices (h±,k), nd symptotes y ± x The Eqution of Hyperol Centered t (h,k) Opening Up nd Down The hyperol hs foci (h,k±c),where ( y k) ( x h) =, vertices (h,k±), nd symptotes y ± x

4 WORKSHEET ( ) 1. Find the vertex, focus, nd directrix for ech prol. ) y = x² +x 5 ) y = x² +4x 1. Find the eqution of the prol determined y the given informtion. Focus (0, -1) nd directrix y 3. Find the vertex, xis of symmetry, x-intercept, y-intercept, focus, nd directrix for the prol: y = 1 x 4 + ( ) 4 3. Mtch the eqution of the ellipse with its grph. ) ( ) ( ) x 1 y ) + y y 3 64 x c) ( x 1 ) + x + 1 y d) ( ) + x 1 5 y 3 49 e) ( ) + x + y f) ( )

5 4. Find the vertices, co-vertices, nd foci of the ellipse. ) x + y () ( x + 1) ( y ) Write the stndrd eqution of the ellipse. Find the center, foci, vertices, nd co- vertices of the ellipse. x + 4y + 8x - 8y + 16 = 0 6. Write the stndrd eqution for the ellipse with the given chrcteristics. ) Center (0,0) Vertex: (0,-3) Co-Vertex: (-,0) ) vertices: (,0) nd (,8) co-vertices: (0,4) nd (4,4) 7. Find the vertices, co-vertices, foci, nd symptotes of the hyperol. ) y x 5 64 ) x y = Find the vertices, co-vertices, nd foci of the hyperol. x + 6 y ) ( ) ) ( y + ) ( x 1) 9 9. Write the eqution in stndrd form. Find the vertices, co-vertices, nd foci of the hyperol. 4x - 9y - 3x - 108y = Write the stndrd eqution for the hyperol with the given chrcteristics. () Foci: (-10, 0) nd (10, 0) co-vertices: (0,-6) nd (0,6) () vertices: (-6,0) nd (6,0) co-vertices: (0,-5) nd (0,5) 11. Write the stndrd eqution for the hyperol with the given chrcteristics. () center: (8,-7) Foci: ( 8, -1) nd ( 8, - ) co-vertices: (5,-7) nd (11,-7) () center: (-4,8) vertices: (-4,3) nd (-4,13) co-vertices: (-11,8) nd (3,8)