Θέματα Αναλυτική Γεωμετρία (Ζαφειρίδου)

ANALUTIKH GEWMETRIA

1. 'Estw d mÐa eujeÐa tou q¸rou, M1 èna shmeÐo thc d kai ~u èna mh mhdenikì

di�nusma par�llhlo proc thn d. Na apodeiqjeÐ ìti h apìstash ρ(M0, d)enoc shmeÐou M0 tou q¸rou apì thn d eÐnai:

ρ(M0, d) =|[~u,

−−−→M1M0]||~u|

2. Na apodeiqjeÐ ìti èna di�nusma ~u = {u1, u2, u3} eÐnai par�llhlo sto epÐpedo

me exÐswsh Ax + By + Cz + D = 0 tìte kai mìnon tìte ìtan

Au1 + Bu2 + Cu3 = 0.

3. (∗) Na prosdioristeÐ h sqetik jèsh twn eujei¸n d1 kai d2 kai na brejeÐ h

apìstash metaxÔ twn, an

d1 :

{x− 3y + z = 0x + y − z + 4 = 0

d2 :

{x = 3 + t

y = −1 + 2tz = 4

4. (∗) Na brejeÐ to shmeÐo summetrikì tou A = (−2, 1, 0) wc proc to epÐpedo

3x− 4y − 10z + 8 = 0.

5. (∗) Na prosdioristeÐ to eÐdoc thc epif�neiac

6x2 + 6z2 + 30z + 6x− 11 = 0.

6. (∗) Na brejeÐ h kanonik exÐswsh thc kampÔlhc:

x2 − 12xy − 4y2 + 12x + 8y + 5 = 0.

7. Na grafoÔn oi parametrikèc exis¸seic tou epipèdou

9x + 2y + 9z = 5

(∗) To sÔsthma suntetagmènwn eÐnai orjokanonikì.

ANALUTIKH GEWMETRIA

1. Na apodeiqjeÐ ìti an ta dianÔsmata ~u1, ~u2, ..., ~un eÐnai grammik¸c anex�rthta

kai

~u = λ1~u1 + λ2~u2 + ... + λn~un,

tìte oi arijmoÐ λ1, λ2,...,λn eÐnai monadikoÐ.

2. (∗) Na apodeiqjeÐ ìti h apìstash ρ(M0, π) enìc shmeÐou M0 = (x0, y0, z0)apì to epÐpedo π : Ax + By + Cz + D = 0 mporeÐ na upologisteÐ apì ton

tÔpo:

ρ(M0, π) =|Ax0 + By0 + Cz0 + D|√

A2 + B2 + C2.

3. (∗) Na prosdioristeÐ h sqetik jèsh twn eujei¸n d1 kai d2 kai na brejeÐ h

apìstash metaxÔ twn, an

d1 :

{x− 3y + z = 0x + y − z + 4 = 0

d2 :

{x = 3 + t

y = −1 + 2tz = 4


5x2 + 4xy + 8y2 − 32x− 56y + 80 = 0


6y2 + 6z2 + 30z + 6y − 11 = 0.

6. (∗) Na brejeÐ h efaptomènh thc kampÔlhc

5x2 + 4xy + 8y2 − 32x− 56y + 80 = 0

sto shmeÐo F = (4, 4).

7. (∗) Na brejeÐ h probol tou A = (−2, 1, 0) sto epÐpedo

3x− 4y − 10z + 8 = 0.


ANALUTIKH GEWMETRIA

1. DÐnontai trÐa mh sunepÐpeda dianÔsmata ~a =−→OA, ~b =

−−→OB kai ~c =

−→OC.

Na apodeiqjeÐ ìti an to sÔsthma O~a~b~c eÐnai jetik¸c prosanatolismèno,

tìte o ìgkoc V tou parallhlepipèdou me akmèc OA, OB, OC isoÔtai me

〈~a,~b,~c〉.

2. Na apodeiqjeÐ ìti an èna epÐpedo Π èqei exÐswsh n1x + n2y + n3z + c = 0wc proc èna orjokanonikì sÔsthma suntetagmènwn Oxyz, tìte to di�nusma

~n = {n1, n2, n3} eÐnai k�jeto sto Π.

3. (∗) Na brejeÐ h probol tou A = (−2, 1, 0) sthn eujeÐa

d :

{x = 1 + t

y = −3− 4tz = −3− 3t


x2 − 12xy − 4y2 + 12x + 8y + 5 = 0.


6y2 + 6z2 + 30z + 6y − 11 = 0

6. (∗) Na brejeÐ to Ôyoc tou parallhlepipèdou me akmèc KA, KB, KC apì

thn koruf A an K = (1, 1, 1), A = (3, 3, 5), B = (2, 3, 1), C = (6, 0, 3).

7. Na prosdioristeÐ h sqetik jèsh twn eujei¸n d1 kai d2

d1 :

{x− 3y + z = 0x + y − z + 4 = 0

d2 :

{x = 3 + t

y = −1 + 2tz = 4


ANALUTIKH GEWMETRIA (18/01/2010)

1. Na apodeiqjeÐ ìti an ta dianÔsmata ~u1, ~u2, ..., ~un, , n ≥ 2, eÐnai grammik¸c exarthmèna kai tadianÔsmata ~u1, ~u2, ..., ~un−1 eÐnai grammik¸c anex�rthta, tìte to di�nusma ~un eÐnai grammikìcsunduasmìc twn dianusm�twn ~u1, ~u2, ..., ~un−1.

2. Na dojeÐ o orismìc tou exwterikoÔ ginomènou ~u× ~v twn eleÔjerwn dianusm�twn ~u kai ~v.

Na apodeiqjeÐ ìti h apìstash ρ(N, ε) enìc shmeÐou N tou q¸rou apì thn eujeÐa ε upologÐzetaiapì ton tÔpo

ρ(N, ε) =|~u×

−−−→NM0||~u|

, ìpou ~u ‖ ε kai M0 ∈ ε.

3. Na apodeiqjeÐ ìti an h tom twn epipèdwn π1 : A1x + B1y + C1z + D1 = 0 kaiπ2 : A2x + B2y + C2z + D2 = 0 eÐnai h eujeÐa ε, tìte to di�nusma

~a =

{∣∣∣∣ B1 C1

B2 C2

∣∣∣∣ ,

∣∣∣∣ C1 A1

C2 A2

∣∣∣∣ ,

∣∣∣∣ A1 B1

A2 B2

∣∣∣∣}eÐnai mh mhdenikì kai par�llhlo sthn ε.

4. (∗) Na apodeiqjeÐ ìti h parabol me exÐswsh y2 = 2px, p > 0, eÐnai to sÔnolo ìlwn twn shmeÐwntou epipèdou, twn opoÐwn h apìstash apì thn estÐa thc parabol c isoÔtai me thn apìstashapì thn dieujetoÔsa thc parabol c.

5. Na brejoun oi parametrikèc exis¸seic kai h kartesian exÐswsh (exÐswsh thc morf cAx + By + Cz + D = 0) tou epipèdoÔ π, to opoÐo eÐnai par�llhlo sto di�nusma −→a = {2, 3, 5}

kai perièqei thn eujeÐa ε :

x = 1 + t

y = −3− 4t

z = −3− 3t

6. (∗) Na brejeÐ h apìstash metaxÔ twn eujei¸n: ε1 :

x = 1 + 2t

y = 2− 2t

z = −t

ε2 :

x = −2− 2t

y = −2 + 3t

z = 4

7. (∗) Na brejeÐ h kanonik exÐswsh kai to kanonikì sÔsthma thc kampÔlhc:

6xy + 8y2 − 12x− 26y + 11 = 0.

8. (∗) Na prosdioristeÐ to eÐdoc thc epif�neiac: 2x2 − 4y2 − 6x + 8y − z + 1 = 0.

9. (∗) Na brejeÐ h exÐswsh tou epipèdou pou perièqei to shmeÐo A = (3,−1, 2) kai eÐnai k�jetosthn eujeÐa

ε :

{x + y − z − 1 = 0

2x− y + 2z + 7 = 0.

(*)− To sÔsthma suntetagmènwn eÐnai orjokanonikì.

BajmologÐa: 1/7 to jèma. Epilèxte 7 apì ta 9 jèmata.

Kal EpituqÐa!


1. 'Estw Π èna epÐpedì, M0 ∈ Π kai −→a ,−→b duo mh suggrammik�, par�llhla proc to Π dianÔsmata.

Na apodeiqjeÐ ìti h apìstash ρ(N, Π) enìc shmeÐou N tou q¸rou apì to Π eÐnai

ρ(N, Π) =| < −→a ,

−→b ,−−−→M0N > |

|[−→a ,−→b ]|

,

ìpou < −→a ,−→b ,−−−→M0N > eÐnai to miktì ginìmeno twn dianusm�twn −→a ,

−→b ,−−−→M0N kai [−→a ,

−→b ] eÐnai

to exwterikì ginìmeno twn dianusm�twn −→a kai−→b .

2. Na apodeiqjeÐ ìti èna di�nusma −→u = {u1, u2, u3} eÐnai par�llhlo proc to epÐpedoΠ : Ax + By + Cz + D = 0 an kai mìnon an Au1 + Bu2 + Cu3 = 0.

3. (∗) Na apodeiqjeÐ ìti h parabol me exÐswsh y2 = 2px, p > 0, eÐnai to sÔnolo ìlwn twn shmeÐwntou epipèdou, twn opoÐwn h apìstash apì thn estÐa F =

(p2, 0

)thc parabol c isoÔtai me thn

apìstash apì thn dieujetoÔsa x = −p2thc parabol c.

4. Na apodeiqjeÐ ìti an ta dianÔsmata ~u1, ~u2, ..., ~un eÐnai grammik¸c exarthmèna an kai mìnon anèna apì ta dianÔsmata aut� eÐnai grammikìc sunduasmìc twn �llwn.

5. Na grafeÐ h exÐswsh tou epipèdou Π pou eÐnai par�llhlo ston �xona Ox kai perièqei thn eujeÐa

ε :

{6x− y + z = 0

5x + 3z − 10 = 0

6. (∗) Na brejeÐ h apìstash metaxÔ twn eujei¸n: ε1 :

x = −3− 3t

y = −2 + 4t

z = 1 + t

ε2 :

x = 1− 6t

y = 1 + 8t

z = 1 + t


5x2 + 6xy + 5y2 − 10x− 6y − 3 = 0.

8. (∗) Na prosdioristeÐ to eÐdoc thc epif�neiac: 3x2 − 2y2 + 12x− 4y − 12z + 34 = 0.

9. (∗) DÐnetai h eujeÐa

ε :

{x− 2y + z − 10 = 0

x− 4y − z − 4 = 0.

(aþ) Na grafoÔn oi parametrikèc exis¸seic thc ε.

(bþ) Na brejeÐ h orjog¸nia probol tou shmeÐou P = (6, 3, 14) sthn eujeÐa ε.

10. DÐnetai to epÐpedo Π : 2x− 3y + 5z − 11 = 0.

(aþ) Na grafoÔn oi parametrikèc exis¸seic thc eujeÐac pou eÐnai k�jeth sto Π kai dièrqetaiapì to shmeÐo N = (−3, 8,−7).

(bþ) Na brejeÐ to shmeÐo shmmetrikì tou N = (−3, 8,−7) wc proc to Π.


BajmologÐa: 1/7 to jèma. Epilèxte 7 apì ta 10 jèmata.


1. Na apodeiqjeÐ ìti an h eujeÐa ε eÐnai eujeÐa tom c twn epipèdwn π1 kai π2, ìpou

π1 :A1x + B1y + C1z + D1 = 0

π2 :A2x + B2y + C2z + D2 = 0

tìte to di�nusma ~a =

{∣∣∣∣ B1 C1

B2 C2

∣∣∣∣ ,

∣∣∣∣ C1 A1

C2 A2

∣∣∣∣ ,

∣∣∣∣ A1 B1

A2 B2

∣∣∣∣} eÐnai mh mhdenikì kai par�llhlo

sthn ε.

2. Na apodeiqjeÐ ìti h apìstash tou shmeÐou N apì thn eujeÐa ε upologÐzetai apì ton tÔpo:

ρ(N, ε) =|−−−→M0N × ~a|

|~a|, ìpou M0 ∈ ε kai ~a ‖ ε.

3. Na apodeiqjoÔn oi prot�seic:

(aþ) An ~u1, ..., ~un, ~un+1 eÐnai grammik¸c anex�rthta, tìte ~u1, ..., ~un eÐnai grammik¸c anex�rthta.

(bþ) An ~u1, ..., ~un eÐnai grammik¸c exarthmèna, tìte ~u1, ..., ~un, ~un+1 eÐnai grammik¸c exarthmèna.

4. (∗) 'Estw K = (1, 1, 1) A = (−1, 1,−1), B = (2, 0, 1) kai C = (2, 3, 5). Na brejeÐ to Ôyoc tou

parallhlepipèdou me akmèc−−→KA,

−−→KB,

−−→KC apì thn koruf C proc thn èdra pou perièqei tic

korufèc K, A, B.

5. (∗) Na brejeÐ h apìstash metaxÔ twn eujei¸n

ε1 :

x = 3

y = 3t

z = 6− t

ε2 :

{x + 2y − z + 1 = 0

2x− 3y + z − 4 = 0

6. (∗) Na brejeÐ h kanonik exÐswsh thc kampÔlhc: 2x2 + 4xy + 5y2 − 6x− 8y − 1 = 0.


8. (∗) DÐnetai to epÐpedo Π : 4x− 3y − 2z − 1 = 0.

(aþ) Na grafoÔn oi parametrikèc exis¸seic tou Π.

(bþ) Na grafoÔn oi parametrikèc exis¸seic thc eujeÐac pou eÐnai k�jeth sto Π kai dièrqetaiapì to shmeÐo N = (6, 2, 12).

9. (∗) Na brejeÐ h orjog¸nia probol tou shmeÐou A = (6, 3, 14) sthn eujeÐa ε :

x = 1 + 3t

y = −2 + t

z = 5− t

10. Na brejeÐ h kartesian exÐswsh tou epipèdoÔ π, to opoÐo perièqei to shmeÐo A = (2, 3, 1) kaithn eujeÐa

ε :

{x + y − z − 1 = 0

2x− y + 2z + 7 = 0.


Epilèxte 8 apì ta 10 jèmata.






π1 :A1x + B1y + C1z + D1 = 0

π2 :A2x + B2y + C2z + D2 = 0


{∣∣∣∣ B1 C1

B2 C2

∣∣∣∣ ,

∣∣∣∣ C1 A1

C2 A2

∣∣∣∣ ,

∣∣∣∣ A1 B1

A2 B2


sthn ε.


ρ(N, ε) =|−−−→M0N × ~a|



ε1 :

x = 3

y = 3t

z = 6− t

ε2 :

{x + 2y − z + 1 = 0

2x− 3y + z − 4 = 0




(aþ) Na grafoÔn oi parametrikèc exis¸seic thc eujeÐac pou eÐnai k�jeth sto Π kai dièrqetaiapì to shmeÐo N = (6, 2, 12).

(bþ) Na grafoÔn oi parametrikèc exis¸seic tou Π.


x = 1 + 3t

y = −2 + t

z = 5− t


ε :

{x + y − z − 1 = 0

2x− y + 2z + 7 = 0.



−−→KB,


korufèc K, A, B.





ε1 :

x = 3

y = 3t

z = 6− t

ε2 :

{x + 2y − z + 1 = 0

2x− 3y + z − 4 = 0




π1 :A1x + B1y + C1z + D1 = 0

π2 :A2x + B2y + C2z + D2 = 0


{∣∣∣∣ B1 C1

B2 C2

∣∣∣∣ ,

∣∣∣∣ C1 A1

C2 A2

∣∣∣∣ ,

∣∣∣∣ A1 B1

A2 B2


sthn ε.


ρ(N, ε) =|−−−→M0N × ~a|







−−→KB,


korufèc K, A, B.


(aþ) Na grafoÔn oi parametrikèc exis¸seic tou Π.

(bþ) Na grafoÔn oi parametrikèc exis¸seic thc eujeÐac pou eÐnai k�jeth sto Π kai dièrqetaiapì to shmeÐo N = (6, 2, 12).


x = 1 + 3t

y = −2 + t

z = 5− t


ε :

{x + y − z − 1 = 0

2x− y + 2z + 7 = 0.



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Θέματα Αναλυτική Γεωμετρία (Ζαφειρίδου)