Mavzu: Aylana tenglamasi va unga doir misollar yechish Reja

2.Aylana tenglamasi va unga doir misollar yechish

To’g’ri chiziq va ikkinchi tartibli egri chiziqlarning tenglamalari berilgan quyidagilar topilsin:

1) Ikkinchi tartibli egri chiziqning barcha elementlari;
2) ikkinchi tartibli egri chiziq bilan to’g’ri chiziqning kesishgan nuqtalari.
Chizmasini chizing. (x+4)²+(y-5)²=16, x-y+5=0
Yechish. a) Markazi M(a;b)nuqtala va radiusi R bo’lgan aylana kanonik tenglamasi (x-a)²+(y-b)²=R² (1) ko’rinishda bo’ladi.
Demak, (x+4)²+(y-5)²=16 tenglama bilan berilganikkinchi tartibli egri chiziq aylana bo’lib, uning markazi M(-4;5) nuqtada,radiusi R esa 4 ga teng. To’g’ri chiziq tenglamasidan y — x + 5 ekanini ko’ramiz. Buni aylana tenglamasiga qo’yib ularning kesishish nuqtasini topamiz:
(x+4)²+(y-5)²=16;
(x+4)²+(x+5-5)²⁼16
x²+8x+16+x²=16; 2x²+8x=0; x²+4x=0; x₁=0, x₂=-4 y=x+5=0+5=5; y²=x²+5=-4+5=1
Demak, kesishish nuqtalari: (0;5) va (-4:1).
1-misol. Markazi (3; -4) nuqtada yotgan hamda radiusi 6 ga tеng bo`lgan aylana tеnglamasini tuzing.
Yechish: Shartga ko`ra =3, b=-4 va R=6.
Bеrilganlarni (4) tеnglamaga qo`yamiz:
(x-3)<sup>2</sup>+(y+4)<sup>2</sup>=6<sup>2</sup> ,
bundan, x<sup>2</sup>-6x+9+y<sup>2</sup>+8y+16=6<sup>2</sup>,
x<sup>2</sup>+y<sup>2</sup>-6x+8y-11=0 .
2-misol. Radiusi 7 va markazi (5; 4) nuqtada bo`lgan aylana tеnglamasini toping.
Yechish: Masala shartiga asosan =5, b=4, R=7.
(4) tеnglamaga asosan:
(x-5)²+(y-4)⁴=7²,
x²-10x+25+y²-8y+16-49=0,
x²+y²-10x-8y-8=0.
Bu izlangan tеnglama.
3-misol. 5x²-10x+5y²+20y-20=0 tеnglama bеrilgan. Aylana markazi va uning radiusini toping.
Yechish: A=5, B=-10, C=20, D=-20 bеrilgan. (8) formulalar yordamida ,b va R² ni topamiz.

dеmak, a=1, v=-2 va R=3

y

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-4
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