Misali na Tambayoyin Tattaunawa kan Tsarin Nazarin Lissafi
Pendahuluan
Tsarin lissafi na nazari wani reshe ne na lissafi wanda ke haɗa algebra da lissafi don magance matsalolin da suka shafi sarari da siffa. Kayan aiki ne mai ƙarfi wanda ke ba mu damar yin nazarin matsalolin lissafi ta amfani da lissafi da daidaitawa. Wannan labarin zai tattauna misalai da yawa na matsalolin yanki na lissafi na nazari kuma ya tattauna su dalla-dalla don sauƙaƙe fahimta mai zurfi.
Misali Tambaya ta 1: Daidaito a Layi
Tambaya:
An ba da maki biyu A(1, 2) da B(3, 7). Kayyade daidaiton layin da ya ratsa waɗannan maki biyu.
Tattaunawa:
Domin nemo lissafin layin da ke ratsa maki biyu, za mu iya amfani da dabarar gradient (slope) m:
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Tare da maki A(x1, y1) = (1, 2) da maki B(x2, y2) = (3, 7):
\[ m = \frac{7 – 2}{3 – 1} = \frac{5}{2} \]
Na gaba, muna amfani da dabarar lissafin layin:
\[ y – y_1 = m(x – x_1) \]
Sauya maki ɗaya, misali aya A(1, 2):
\[ y – 2 = \frac{5}{2}(x – 1) \]
Maida wannan fom zuwa lissafi mai bayyana don y:
\[ y – 2 = \frac{5}{2}x – \frac{5}{2} \]
\[ y = \frac{5}{2}x – \frac{5}{2} + 2 \]
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Don haka, daidaiton layin shine:
\[ y = \frac{5}{2}x – \frac{1}{2} \]
Misali Tambaya ta 2: Da'ira
Tambaya:
Kayyade lissafin da'irar da ke tsakiya a wurin C(-2, 3) kuma tare da radius na 4.
Tattaunawa:
Daidaiton da'ira mai tsakiya a (h, k) da radius r shine:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Daga tambayar, tsakiyar da'irar (h, k) = (-2, 3) da radius r = 4. Don haka,
\[ (x + 2)^2 + (y - 3)^2 = 4^2 \]
\[ (x + 2)^2 + (y - 3)^2 = 16 \]
Don haka, daidaiton da'irar shine:
\[ (x + 2)^2 + (y - 3)^2 = 16 \]
Misali Tambaya ta 3: Parabola
Tambaya:
Kayyade daidaiton parabola mai tsaye tare da karkata a (1, -2) kuma mayar da hankali a (1, 0).
Tattaunawa:
Ga parabola mai tsaye tare da vertex (h, k), lissafin gabaɗaya shine:
\[ (x – h)^2 = 4p(y – k) \]
Idan aka yi la'akari da karkata (h, k) = (1, -2), muna buƙatar nemo ƙimar p. Mayar da hankali kan parabola shine (h, k + p), kuma daga matsalar, ma'anar ma'anar ita ce (1, 0):
\[k + p = 0 – (-2) = 2 \]
Don haka:
\[ p = 2 \]
Don haka, jimlar lissafi ta zama:
\[ (x – 1)^2 = 4 \cdot 2 (y + 2) \]
\[ (x – 1)^2 = 8(y + 2) \]
Don haka, lissafin parabola shine:
\[ (x – 1)^2 = 8(y + 2) \]
Misali Tambaya ta 4: Ellipse
Tambaya:
An ba da ellipse mai tsakiya a wurin (0, 0), tsawon babban 10 da ƙaramin 6. Kayyade daidaiton ellipse.
Tattaunawa:
Tsakiyar ellipse (h, k) shine (0, 0), tsawon babban ellipse 2a = 10 don haka a = 5, da tsawon ƙaramin ellipse 2b = 6 don haka b = 3. Daidaito na gaba ɗaya don ellipse mai tsakiya a (0, 0) shine:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Sauya dabi'un a da b:
\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \]
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Saboda haka, lissafin ellipse shine kamar haka:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Misali Tambaya ta 5: Hyperbola
Tambaya:
Idan aka yi la'akari da hyperbola mai tsakiya a (1, -3), tsawon axis mai juyawa shine 8 kuma tsawon axis mai haɗin kai shine 6. Ka ƙayyade daidaiton hyperbola.
Tattaunawa:
Ga hyperbola mai tsakiya (h, k) da kuma axis na kwance, lissafin gabaɗaya shine:
\[ \frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1 \]
Tsakiyar hyperbola (h, k) shine (1, -3), tsawon axis mai wucewa shine 2a = 8 don haka a = 4, kuma tsawon axis mai haɗuwa shine 2b = 6 don haka b = 3. Saboda haka, daidaiton hyperbola shine:
\[ \frac{(x – 1)^2}{4^2} – \frac{(y + 3)^2}{3^2} = 1 \]
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Don haka, lissafin hyperbola shine:
\[ \frac{(x – 1)^2}{16} – \frac{(y + 3)^2}{9} = 1 \]
Kammalawa
Tsarin lissafi na nazari hanya ce mai ƙarfi don nazarin siffofi da tsare-tsare na lissafi ta amfani da lissafin algebra. Ta hanyar fahimtar mahimman ra'ayoyi na lissafi na layi, da'ira, parabolas, ellipses, da hyperbolas, za mu iya magance matsaloli daban-daban na lissafi cikin sauƙi. Wannan labarin yana ba da misalai da tattaunawa kan muhimman matsaloli a cikin lissafi na nazari don taimakawa zurfafa fahimtar ku. Ƙarin matsalolin aiki na iya taimakawa wajen ƙarfafawa da faɗaɗa fahimtar ku game da wannan kayan.
