Rezolvari Geometrie
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SETUL I 1) t: ex + y -2 + e^2=0 n: x-ey+e=0 2) t: x-y-1=0 n:x+y-1=0 3) t: y-3=0 n: x=0 4) MT=15 2 /7 MN=15sqrt(2) PT=3/7 PN=21 5) A(1,0) K1=K2=1/2sqrt(2) 6) x2+(y-(a2-b2))2= b4/a2 7) x=3at/(1+t^3) y=3at^2/(1+t^3) 8) R=4y5/4 9) R=2S 10)R=(m/m+1)*pm-1/m 11)ds=(sin O/m)m-1dO 12)ds=2a cos(alfa/2)dalfa 13)ds=chx 14)tgV=p/p' 15)t:Y-y=(p'tgO+p)/(p'-ptgO)*(X-x) n: Y-y=(ptgO-p')/(p'tgO+p)*(X-x) 16)1/k 17)PT=-2x PN=p 18)|MT|=1 19)MT=1/k*sqrt(1+k^2p) MN=k*sqrt(1+k^2p) PT=1/k*p PN=kp 20)x=2asin2(fi) y=2a(sin3(fi))/cos(fi) 21)x(x^2+y^2)-2ay^2=0 22)x=a(t^2-1)/t^2+1 y=at(t^2-1)/t^2+1 23)A(0,-3) B(-18,6) C(-2,2) 24)y=x y=49/9x 25)y=(7x-5)/(5x-3) 26)y=x y=x-4 27)x+2y-3=0 2x-y-1=0 28)16x-y-5=0 2x+32y-97=0 29)t: x-7y-9=0 n: 7x+y-13=0 30)MT=sqrt(5)/2 MN=sqrt(5) PT=1/2 PN=2 31) MT=sqrt(2) PT=1 MN=sqrt(2) PN=1 32)MT=sqrt(2) PT=1 MN=sqrt(2) PN=1 33)PT= 1/|b| 34)y=aebx 35)R=|a|ch2x 36)R=4asin(t/2) 37)R/MN=2 38)K=4sqrt(2) R=sqrt(2)/8 39)k=(p^2+p'^2-pp'')/(p^2+p'^2)3/2 R=(p^2+p'^2)3/2/|p^2+p'^2-pp'| 40)K=R=1 41)K=4sqrt(2)/3*|a| R=3sqrt(2)/8a 42)PT=2asin(2t/2)tg(t/2) PN=asint 43)Y-y=-my/nx*(X-x) 44)R=St/Sn 45)S=(n+1)R 46)A(sqrt(-p/3),0) B(-sqrt(-p/3),0) 47)punct izolat pt orice p>0 48)M(1,0) y=x-1 y=1-x 49)(F''xy)^2-F''xxF''yy<0 in M 50)O(0,0) de tip nod 51)A(a,0) nod pt. Curba © if a>b 52)O(0,0) pct dublu 53)p>0 A(sqrt(p/3),0) B(sqrt(p/3),0) pct izolate
54)O(0,0) izolat 55)M(1,0) pct dublu, dreptele tangente: y=x-1 y=1-x 56)doar conicele degenerate admit puncte singulare 57)AB concav 58)concava spre pol 59)x=a(sint+cost) y=a(tsint+cost) 60)x=t-ath(t/a) y=a/ch(t/a) 61)1/R=1/4asin(t/2) 62)S*1/R=const 63)??? 64)cercurile sunt singurele curbe plane avand curbura constanta 65)y=ach(x/a) 66)r=3 67)r=1 68)(x-alfa)^2+(y-beta)^2=r^2 alfa=x-y'(x'^2+y'^2)/(x'y''-x''y') beta=y+x'(x'^2+y'^2)/(x'y''-y'x'') 69)alfa=x-y'^2(1+y'2)/y'' beta=y+y'^2(1+y'^2)/y'' r=(1+y'^2)^3/2/|y''| 70)(x-a)^2+(y-beta)^2=r^2 alfa=x-ash^2(x/a) beta=y+ach(x/a) r=ach^2(x/a) 71)y=C/2-1/2C*x^2 72)y-x=-4/27 73)x^2/3 + y^2/3 =k^2/3 74)---75)y^2=2px 76)x=x-y'*(1+y^2)/y'' y=y-(1+y'^2)/y'' 77)x=c^2/a* cos^3(t) y=-c/b*sin^3(t) 78)y^2=8/27(x-p)^3 79)x=x-x'(k-s) y=y+y'(k-s) 80)x=asint+(k-at)cost y=acost-(k-at)sint 81)x=t-2shtcht y=4cht+2ch^2t 82)x=t+(k-1)sht y=cht+(1-k)tht
SETUL II 1) x=rcos^2(t) y=rsin(t)cos(t) z=rsin(t) 2) x^2 + y^2 +z^2 -r^2=0 x^2 +y^2 -rx=0 3) elice circulara 4) --5) x=atcos(t) y=atsin(t) z=bt 6) elice conica 7) planul este tangent la curba in M(1,0,-2) 8) ds=sqrt(a^2 +k^2)dt 9) l=pi/2sqrt(a^2 +k^2) 10)l=3 11)ds=a|cos(t)|dt 12)A(2,0,4) B(3,1,2) 13)-7/2 14)curba este tangenta la plan 15)x-acos(t)/asin(t)=y-asin(t)/acos(t)=z-bt/b 16)-asin(t)(X-acos(t)) + acos(t)(Y-bsin(t)) +b(Z-bt)=0 17)X-x/sqrt(a)=Y-y/sqrt(b)=Z-z/sqrt(2z) 18)cos(alfa)=-(asin(t))/sqrt(a^2+b^2) cos(beta)=acos(t)/sqrt(a^2+b^2) cos(fi)=-b/sqrt(a^2+b^2) 19)x=(t^2-1)/(t^2+t+1) y=(2t+1)/(t^2+t+1) z=-t(t+2)/(t^2+t+1) 20)x+10y-8z+27=0 21)3x+2y-z=0
22)lambda=1 3x+2y-z-1=0 23)(x-1)/1=(y-0)/ 1=(z-1)/1 n: 1(x-1)+1(y-0)+1(z-1)=0 24)(x-1)/2=(y-1)/-1=(z+1)/0 n: 2x-y-1=0 25)M(0,2,1) 26)x/1=y-2/2=z-1/2 n: x+2y+2z-6=0 27)y=3x^2 z=0 28)lipsa 29)x-sqrt(3)/-1= y-1/sqrt(3)=z-1/sqrt(3) n: x-sqrt(3)y-sqrt(3)z+sqrt(3)=0 30)lipsa 31)x-y-z+3=0 32)x=t+1 y=t^2+t+2 z=2-t^2 33)bsintX-bcostY+aZ-abt=0 34)aX-Z=0 35)x/0=y-a/ab+a^2=2z-bpi/0 36)A=ab/t[asin(alnt)-cos(alnt)] B=ab/t[acos(alnt)+asin(alnt)] C=a/t(1+a^2) 37)delta=-asinO/sqrt(a^2+b^2)*i + acosO/sqrt(a^2+b^2)*j +bk b=bsinO/sqrt(a^2+b^2)*i -bcosO/sqrt(a^2+b^2) +ak/sqrt(a^2+b^2) n=-cosOi +sinOj 38)aj+bk/sqrt(a^2+b^2) -bj+ak/sqrt(a^2+b^2) -i 39)formule 40)1/R=a/a^2+b^2 41)1/T=b/a^2+b^2 42)T/R=a/b 43)1/R=asqrt(1+a^2)/(1+a^2+b^2)t 44)1/T=ab/(1+a^2+b^2)*t 45)elice 46)cosV=a/sqrt(1+a^2+b^2) 47)paralel cu planul xOy 48)A=-ab^2 B=-ab^2 C=(1+a^2)b 49)1/T=(1+t^2/1+t^2+t^4)^3/2 50)delta=2/3*i +2/3*j +1/3*k t: x-2/2=y-1/2=z/1 51)??? 52)x-2/2=y-1/-1=z/-2 53)x=ncos(t) y=rsin(t)cos(t) z=rsin(t) 54)elice circulara 55)segm AB cu a(r,0,0) B(0,-r,0) 56)cercul x^2+y^2-r^2=0 z=0 57)ds=sqrt(a^2+t^2+b^2)dt 58)x^2/a^2 + y^2/b^2 -z^2/b^2=o z=barctg(y/x) 59)ds=sqrt(a^2+k^2) 60)l=pi(sqrt(a^2+k^2)) 61)z+2(x-1)^2-2=0 x-y-z+3=0 62)x-y-z-3=0 63)(X-x)+(2t+1)(Y-y)-2t(Z-z)=0 64)X-tcos(alnt)/ab/t*[absin(alnt)-cos(alnt)]= Ytsin(alnT)/ab/t*[acos(alnt)+asin(alnt)]=Z-bt/t(1-a^2) 65)aj+bk/sqrt(a^2+b^2) -bj+ak/sqrt(a^2+b^2) b=-i 66)lipsa 67)R/T=-1 68)2x-y+2z-3=0 69)x-sqrt(2)/-sqrt(2)=y-sqrt(2)/sqrt(2)=z-pi/4 -sqrt(2)x +sqrt(2)y +4z-4pi=0 70)un singur vector de poz r=i+j+k
Setul 4 1. (t): 3x - y - 2 = 0
(n): x + 3y - 4 = 0 2. (t): x - y = 0 (n): x + y = 0 3. (t): 4x - 2y - pi + 2 = 0 (n): 4x + 8y - pi + 8 = 0 4. (t): x + y - 3a = 0 (n): x - y = 0 5. (t): 4x - 2y - a = 0 (n): 2x + 4y - 3a = 0 6. (t): x - a*sqrt(2) = 0 (se citeste "x minus a radical din 2 egal 0") (n): y = 0 7. (t): x - 7y - 9 = 0 (problema 29 din setul vechi, varianta "d") (n): 7x + y - 13 = 0 8. (t): 5x + y - 5 = 0 (n): x - 5y - 1 = 0 9. (t): x - y - 1 = 0 (n): x + y - 1 = 0 10.(t): 2x - 5y + 17 = 0 (n): 5x + 2y - 30 = 0 11.(t): 2x - y + 4 = 0 (n): x + 2y - 3 = 0 12.(t): x + y - 1 = 0 (n): x - y = 0 13.(t): x + y - 1 = 0 (n): x - y - 1 = 0 14.(t): 2bx + 2ax - ab*sqrt(2) = 0 (n): 4ax - 4by - (a^2 - b^2)*sqrt(2) = 0 25. Intersectia cu Ox este punctul A(1,0) R = 5*sqrt(10)/2 26. R = 125/16 27. R = (pi^2)/4 28. R = 2*sqrt(2)/pi 29. R = (4 + pi^2)^2/(8*|a|*pi) 30. urata :)) 31. y^2 - 4xy - 8x = 0 32. x^2 - 4y = 0 33. y^2 - 2px = 0 (problema 75 din setul vechi, varianta "c") 34. y^2 - 1 = 0 35. x^2 - 3y^2 = 0 36. y^2 - 4x - 4 = 0 37. X = - x^3/4 Y = (3x^2 + 8)/4 38. X = at Y=0 39. X = x Y = x^2
54)O(0,0) izolat 55)M(1,0) pct dublu, dreptele tangente: y=x-1 y=1-x 56)doar conicele degenerate admit puncte singulare 57)AB concav 58)concava spre pol 59)x=a(sint+cost) y=a(tsint+cost) 60)x=t-ath(t/a) y=a/ch(t/a) 61)1/R=1/4asin(t/2) 62)S*1/R=const 63)??? 64)cercurile sunt singurele curbe plane avand curbura constanta 65)y=ach(x/a) 66)r=3 67)r=1 68)(x-alfa)^2+(y-beta)^2=r^2 alfa=x-y'(x'^2+y'^2)/(x'y''-x''y') beta=y+x'(x'^2+y'^2)/(x'y''-y'x'') 69)alfa=x-y'^2(1+y'2)/y'' beta=y+y'^2(1+y'^2)/y'' r=(1+y'^2)^3/2/|y''| 70)(x-a)^2+(y-beta)^2=r^2 alfa=x-ash^2(x/a) beta=y+ach(x/a) r=ach^2(x/a) 71)y=C/2-1/2C*x^2 72)y-x=-4/27 73)x^2/3 + y^2/3 =k^2/3 74)---75)y^2=2px 76)x=x-y'*(1+y^2)/y'' y=y-(1+y'^2)/y'' 77)x=c^2/a* cos^3(t) y=-c/b*sin^3(t) 78)y^2=8/27(x-p)^3 79)x=x-x'(k-s) y=y+y'(k-s) 80)x=asint+(k-at)cost y=acost-(k-at)sint 81)x=t-2shtcht y=4cht+2ch^2t 82)x=t+(k-1)sht y=cht+(1-k)tht
SETUL II 1) x=rcos^2(t) y=rsin(t)cos(t) z=rsin(t) 2) x^2 + y^2 +z^2 -r^2=0 x^2 +y^2 -rx=0 3) elice circulara 4) --5) x=atcos(t) y=atsin(t) z=bt 6) elice conica 7) planul este tangent la curba in M(1,0,-2) 8) ds=sqrt(a^2 +k^2)dt 9) l=pi/2sqrt(a^2 +k^2) 10)l=3 11)ds=a|cos(t)|dt 12)A(2,0,4) B(3,1,2) 13)-7/2 14)curba este tangenta la plan 15)x-acos(t)/asin(t)=y-asin(t)/acos(t)=z-bt/b 16)-asin(t)(X-acos(t)) + acos(t)(Y-bsin(t)) +b(Z-bt)=0 17)X-x/sqrt(a)=Y-y/sqrt(b)=Z-z/sqrt(2z) 18)cos(alfa)=-(asin(t))/sqrt(a^2+b^2) cos(beta)=acos(t)/sqrt(a^2+b^2) cos(fi)=-b/sqrt(a^2+b^2) 19)x=(t^2-1)/(t^2+t+1) y=(2t+1)/(t^2+t+1) z=-t(t+2)/(t^2+t+1) 20)x+10y-8z+27=0 21)3x+2y-z=0
22)lambda=1 3x+2y-z-1=0 23)(x-1)/1=(y-0)/ 1=(z-1)/1 n: 1(x-1)+1(y-0)+1(z-1)=0 24)(x-1)/2=(y-1)/-1=(z+1)/0 n: 2x-y-1=0 25)M(0,2,1) 26)x/1=y-2/2=z-1/2 n: x+2y+2z-6=0 27)y=3x^2 z=0 28)lipsa 29)x-sqrt(3)/-1= y-1/sqrt(3)=z-1/sqrt(3) n: x-sqrt(3)y-sqrt(3)z+sqrt(3)=0 30)lipsa 31)x-y-z+3=0 32)x=t+1 y=t^2+t+2 z=2-t^2 33)bsintX-bcostY+aZ-abt=0 34)aX-Z=0 35)x/0=y-a/ab+a^2=2z-bpi/0 36)A=ab/t[asin(alnt)-cos(alnt)] B=ab/t[acos(alnt)+asin(alnt)] C=a/t(1+a^2) 37)delta=-asinO/sqrt(a^2+b^2)*i + acosO/sqrt(a^2+b^2)*j +bk b=bsinO/sqrt(a^2+b^2)*i -bcosO/sqrt(a^2+b^2) +ak/sqrt(a^2+b^2) n=-cosOi +sinOj 38)aj+bk/sqrt(a^2+b^2) -bj+ak/sqrt(a^2+b^2) -i 39)formule 40)1/R=a/a^2+b^2 41)1/T=b/a^2+b^2 42)T/R=a/b 43)1/R=asqrt(1+a^2)/(1+a^2+b^2)t 44)1/T=ab/(1+a^2+b^2)*t 45)elice 46)cosV=a/sqrt(1+a^2+b^2) 47)paralel cu planul xOy 48)A=-ab^2 B=-ab^2 C=(1+a^2)b 49)1/T=(1+t^2/1+t^2+t^4)^3/2 50)delta=2/3*i +2/3*j +1/3*k t: x-2/2=y-1/2=z/1 51)??? 52)x-2/2=y-1/-1=z/-2 53)x=ncos(t) y=rsin(t)cos(t) z=rsin(t) 54)elice circulara 55)segm AB cu a(r,0,0) B(0,-r,0) 56)cercul x^2+y^2-r^2=0 z=0 57)ds=sqrt(a^2+t^2+b^2)dt 58)x^2/a^2 + y^2/b^2 -z^2/b^2=o z=barctg(y/x) 59)ds=sqrt(a^2+k^2) 60)l=pi(sqrt(a^2+k^2)) 61)z+2(x-1)^2-2=0 x-y-z+3=0 62)x-y-z-3=0 63)(X-x)+(2t+1)(Y-y)-2t(Z-z)=0 64)X-tcos(alnt)/ab/t*[absin(alnt)-cos(alnt)]= Ytsin(alnT)/ab/t*[acos(alnt)+asin(alnt)]=Z-bt/t(1-a^2) 65)aj+bk/sqrt(a^2+b^2) -bj+ak/sqrt(a^2+b^2) b=-i 66)lipsa 67)R/T=-1 68)2x-y+2z-3=0 69)x-sqrt(2)/-sqrt(2)=y-sqrt(2)/sqrt(2)=z-pi/4 -sqrt(2)x +sqrt(2)y +4z-4pi=0 70)un singur vector de poz r=i+j+k
Setul 4 1. (t): 3x - y - 2 = 0
(n): x + 3y - 4 = 0 2. (t): x - y = 0 (n): x + y = 0 3. (t): 4x - 2y - pi + 2 = 0 (n): 4x + 8y - pi + 8 = 0 4. (t): x + y - 3a = 0 (n): x - y = 0 5. (t): 4x - 2y - a = 0 (n): 2x + 4y - 3a = 0 6. (t): x - a*sqrt(2) = 0 (se citeste "x minus a radical din 2 egal 0") (n): y = 0 7. (t): x - 7y - 9 = 0 (problema 29 din setul vechi, varianta "d") (n): 7x + y - 13 = 0 8. (t): 5x + y - 5 = 0 (n): x - 5y - 1 = 0 9. (t): x - y - 1 = 0 (n): x + y - 1 = 0 10.(t): 2x - 5y + 17 = 0 (n): 5x + 2y - 30 = 0 11.(t): 2x - y + 4 = 0 (n): x + 2y - 3 = 0 12.(t): x + y - 1 = 0 (n): x - y = 0 13.(t): x + y - 1 = 0 (n): x - y - 1 = 0 14.(t): 2bx + 2ax - ab*sqrt(2) = 0 (n): 4ax - 4by - (a^2 - b^2)*sqrt(2) = 0 25. Intersectia cu Ox este punctul A(1,0) R = 5*sqrt(10)/2 26. R = 125/16 27. R = (pi^2)/4 28. R = 2*sqrt(2)/pi 29. R = (4 + pi^2)^2/(8*|a|*pi) 30. urata :)) 31. y^2 - 4xy - 8x = 0 32. x^2 - 4y = 0 33. y^2 - 2px = 0 (problema 75 din setul vechi, varianta "c") 34. y^2 - 1 = 0 35. x^2 - 3y^2 = 0 36. y^2 - 4x - 4 = 0 37. X = - x^3/4 Y = (3x^2 + 8)/4 38. X = at Y=0 39. X = x Y = x^2
