Mohlala oa Lipotso tsa Puisano tsa Jeometri ea Tlhahlobo
Pendahuluan
Jeometri ea tlhahlobo ke lekala la lipalo le kopanyang algebra le jeometri ho rarolla mathata a amanang le sebaka le sebopeho. Ke sesebelisoa se matla se re lumellang ho sekaseka mathata a jeometri ho sebelisa li-equation le li-coordinate. Sengoloa sena se tla tšohla mehlala e 'maloa ea mathata a sebaka sa jeometri ea tlhahlobo le ho a tšohla ka botlalo ho nolofatsa kutloisiso e tebileng.
Mohlala oa Potso ea 1: Tekanyo ea Mola
Potso:
Fana ka lintlha tse peli A(1, 2) le B(3, 7). Fumana equation ea mola o fetang lintlheng tsena tse peli.
Puisano:
Ho fumana equation ea mola o fetang lintlheng tse peli, re ka sebelisa foromo ea gradient (slope) m:
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Ka ntlha A(x1, y1) = (1, 2) le ntlha B(x2, y2) = (3, 7):
\[ m = \frac{7 – 2}{3 – 1} = \frac{5}{2} \]
Ka mor'a moo, re sebelisa foromo bakeng sa equation ea mola:
\[ y – y_1 = m(x – x_1) \]
Phetolo ea ntlha e le 'ngoe, mohlala ntlha A(1, 2):
\[ y – 2 = \frac{5}{2}(x – 1) \]
Fetolela foromo ena ho equation e hlakileng bakeng sa y:
\[ y – 2 = \frac{5}{2}x – \frac{5}{2} \]
\[ y = \frac{5}{2}x – \frac{5}{2} + 2 \]
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Kahoo, equation ea mola ke:
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Mohlala oa Potso ea 2: Selikalikoe
Potso:
Fumana equation ea selikalikoe se bohareng ba ntlha C(-2, 3) le ka radius ea 4.
Puisano:
Tekanyo e akaretsang ea selikalikoe se nang le setsi ho (h, k) le radius r ke:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Ho tsoa potsong, setsi sa selikalikoe (h, k) = (-2, 3) le radius r = 4. Kahoo,
\[ (x + 2)^2 + (y – 3)^2 = 4^2 \]
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Kahoo, equation ea selikalikoe ke:
\[ (x + 2)^2 + (y – 3)^2 = 16 \]
Mohlala oa Potso ea 3: Parabola
Potso:
Fumana equation ea parabola e otlolohileng ka vertex ho (1, -2) 'me u tsepamise maikutlo ho (1, 0).
Puisano:
Bakeng sa parabola e otlolohileng e nang le vertex (h, k), equation e akaretsang ke:
\[(x – h)^2 = 4p(y – k) \]
Ka lebaka la vertex (h, k) = (1, -2), re hloka ho fumana boleng ba p. Sepheo sa parabola ke (h, k + p), mme ho tsoa bothateng sepheo ke (1, 0):
\[ k + p = 0 – (-2) = 2 \]
E le hore:
\[p = 2 \]
Kahoo, equation e akaretsang e ba:
\[(x – 1)^2 = 4 \cdot 2 (y + 2) \]
\[(x – 1)^2 = 8(y + 2) \]
Kahoo, equation ea parabola ke:
\[(x – 1)^2 = 8(y + 2) \]
Mohlala oa Potso ea 4: Ellipse
Potso:
Ha ho fanoe ka ellipse e nang le bohareng ntlheng (0, 0), bolelele ba axis e kholo 10 le axis e nyane 6. Fumana equation ea ellipse.
Puisano:
Bohareng ba ellipse (h, k) ke (0, 0), bolelele ba axis e kholo ke 2a = 10 e le hore a = 5, 'me bolelele ba axis e nyane ke 2b = 6 e le hore b = 3. Tekanyo e akaretsang ea ellipse e nang le setsi ho (0, 0) ke:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Kenya boleng ba a le b sebakeng sa bona:
\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \]
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Kahoo, equation ea ellipse ke:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Mohlala oa Potso ea 5: Hyperbola
Potso:
Ha ho fanoe ka hyperbola e nang le setsi ho (1, -3), bolelele ba axis e potileng ke 8 mme bolelele ba axis e kopaneng ke 6. Fumana equation ea hyperbola.
Puisano:
Bakeng sa hyperbola e nang le setsi (h, k) le axis e tshekaletseng e rapameng, equation e akaretsang ke:
\[ \frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1 \]
Bohareng ba hyperbola (h, k) ke (1, -3), bolelele ba axis e potileng ke 2a = 8 hoo a = 4, mme bolelele ba axis e kopaneng ke 2b = 6 hoo b = 3. Ka hona, equation ya hyperbola ke:
\[ \frac{(x – 1)^2}{4^2} – \frac{(y + 3)^2}{3^2} = 1 \]
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Kahoo, equation ea hyperbola ke:
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Qetello
Jeometri ea tlhahlobo ke mokhoa o matla oa ho sekaseka libopeho le meaho ea jeometri ho sebelisoa li-equation tsa algebra. Ka ho utloisisa likhopolo tsa motheo tsa li-equation tsa mela, li-circle, li-parabola, li-ellipses le li-hyperbola, re ka rarolla mathata a fapaneng a jeometri habonolo. Sengoloa sena se fana ka mehlala le lipuisano tsa mathata a bohlokoa jeometring ea tlhahlobo ho thusa ho tebisa kutloisiso ea hau. Mathata a mang a ho itloaetsa a ka thusa ho matlafatsa le ho holisa kutloisiso ea hau ea thepa ena.
