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a)
{ (√8 - x)(√8 + x) + x² = 8y
{(y - 2x)(y + 2x) = (y + 1)² - x(4x + 2)
{ 8 - x² +x² = 8y
{ y² - 4x² = y² + 2y +1 - 4x² - 2x
{ 8y = 8 /:8
{ 0 = 2y + 1 - 2x
{ y = 1
{ 0 = 2 + 1 - 2x
{ 2x = 3 /:2
{ y = 1
{ x = 1,5
{ y = 1
b)
{ (x + 2)² - (y + 2)² = 3 + (x - 3)² - y²
{ x - 5y + 3(2x + 3)/2 - (1,5 + 3y)/3 = 0 / *6
{ x² + 4x + 4 - y² - 4y - 4 = 3 + x² - 6x + 9 - y²
{ 6x - 30y + 9(2x + 3) - 2(1,5 + 3y) = 0
{ 4x -4y = - 6 x + 12
{ 6x - 30y + 18x + 27 - 3 - 6y = 0
{ 4y = 10x - 12 /:4
{ 24x - 36y + 24 = 0 / :12
{ y = 2,5x - 3
{ 2x - 3y + 2 = 0
{ y = 2,5x - 3
{ 2x - 3(2,5x - 3) +2 = 0
{ 2x - 7,5x + 9 +2 = 0
{ y = 2,5x - 3
{ 5,5x = 11 /:5,5
{ y = 2,5x - 3
{ x = 2
{ y = 2,5x - 3
{ x = 2
{ y = 2,5*2 - 3
{ x = 2
{ y = 5 - 3
{ x = 2
{ y = 2
Zad. 2)
Dane:
A = (3, - 3)
B = (- 3, 1)
y = ax +b = ?
{ 3a + b = - 3
{ -3a + b = 1
2b = - 2 /:2
b = - 1
3a - 1 = - 3
3a = - 3 + 1
3a = - 2/ :3
a = - 2/3
y = - 2/3x - 1
Odp: Równanie prostej ma postać : y = - 2/3x - 1.
(√8 - x)(√8 + x) + x² = 8y
(y - 2x)(y + 2x) = (y + 1)² - x(4x + 2)
8 - x² + x² = 8y
y² - 4x² = y² + 2y - 4x² - 2x
8 = 8y
y² - y² - 4x² + 4x² = 2y - 2x
y = 8/8 = 1
0 = 2y - 2x
y = 1
- 2y = - 2x
y = 1
x = y
x = 1
odp
x = 1
y = 1
c)
(x + 2)² - (y + 2)² = 3 + (x - 3)² - y²
x - 5y + 3 razy (2x + 3)/2 - (1½ + 3y)/3 = 0
x² + 4x + 4 - y² - 4y - 4 = 3 + x² - 6x + 9 - y²
x - 5y + (6x + 9)/2 - (1½ + 3y)/3 = 0 mnożymy całość przez 6
x² - x² - y² + y² + 4x + 6x - 4y = 3 - 4 + 4 + 9
6x - 30y + 18x + 27 - 3 - 6y = 0
10x - 4y = 12
24x - 36y = 3 - 27
2(5x - 2y) = 12
24x - 36y =- 24
5x - 2y = 6
2x - 3y = - 2
5x = 6 + 2y stąd x = (6 + 2y)/5 wstawiamy do drugiego równania
2(6 + 2y)/5 - 3y = - 2 mnożymy wszystko przez 5
2(6 + 2y) - 15y = - 10
12 + 4y - 15y = - 10
- 11y = - 10 - 12
- 11y = - 22
y = 22/11 = 2
5x - 2y = 6
5x - 2 razy 2 = 6
5x = 6 + 4 = 10
x = 10/5 = 2
odp
x = 2
y = 2
Równanie prostej przechodzącej przez 2 punkty ma postać
Y - YA = (YB - YA)(X - XA)/(XB - XA)
XA = 3
XB = - 3
YA = - 3
YB = 1
Y - 3 = (1 + 3)(X - 3)/( - 3 - 3)
Y - 3 = (4X - 12)/- 6
-6Y + 18 = 4x - 12
- 6y = 4x - 12 - 18
6y = - 4x + 30
y = (- 4x + 30)/6 = - 2(2x - 15)/6 = (-2x + 15)/3 = - 2/3x + 5
równanie prostej ma postać
y = - ⅔x + 5