Ajụjụ na Mkparịta ụka Ihe Nlereanya nke Gburugburu na Tangents
Isi okwu abụọ a na-atụlekarị na mgbakọ na mwepụ, ọkachasị n'ọkwa ụlọ akwụkwọ sekọndrị. Ịghọta echiche na itinye tangents n'ọrụ na okirikiri dị oke mkpa maka ime ka ihe ọmụma gị gbasara geometry dịkwuo omimi. Isiokwu a ga-enye ihe atụ nke nsogbu na mkparịta ụka gbasara okirikiri na tangents iji nye ndị na-agụ akwụkwọ nghọta miri emi.
Okwu Mmalite nke Ozizi nke Gburugburu na Tangents
Lingkaran
Okirikiri bụ otu isi ihe dị n'ime ụgbọelu nke dị anya site na ebe a kapịrị ọnụ nke a na-akpọ etiti okirikiri ahụ. A maara anya a kapịrị ọnụ dị ka radius nke okirikiri ahụ. N'ime mgbakọ na mwepụ, enwere ike ịkọwa okirikiri site na nha nhata:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
ebe \((a, b)\) bụ nhazi nke etiti okirikiri ahụ na \(r\) bụ radius.
Tangent
Tangent na okirikiri bụ ahịrị nke metụrụ okirikiri ahụ kpọmkwem n'otu ebe. A na-akpọ isi ihe a ebe nke tangency. Isi ihe e ji mara tangent bụ na ọ na-adabere na radius nke e si n'etiti okirikiri ahụ ruo na isi ihe nke tangency dọpụta.
Ajụjụ na Mkparịta ụka Ihe Nlereanya
Ajụjụ nke 1: Ịchọpụta Nha nhata nke Ahịrị Tangent
Ajụjụ:
E nyere okirikiri nwere etiti na \( (2, 3) \) na radius 5. Chọpụta nha nhata nke ahịrị tangent na okirikiri ahụ na isi \( P \) yana nhazi \( (5, 7) \).
Azịza:
Nzọụkwụ nke 1: Jide n'aka na isi ihe \( P \) dị n'ime okirikiri ahụ.
Iji lelee ma \(P (5, 7) \) ọ dị na okirikiri nwere etiti \( (2, 3) \) na radius \( 5 \), dochie nhazi nke \(P \) n'ime nha nha nke okirikiri ahụ:
\[ (x – 2)^2 + (y – 3)^2 = 5^2 \]
\[ (5 – 2)^2 + (7 – 3)^2 = 25 \]
\[ 3^2 + 4^2 = 25 \]
\[ 9 + 16 = 25 \]
Ebe ọ bụ na nha nhata ahụ bụ eziokwu, isi ihe dị n'okpuru okirikiri ahụ bụ \(P \).
Nzọụkwụ nke 2: Chọpụta gradient nke radius na-agafe \( (2, 3) \) na \( (5, 7) \):
\[ m_{radius} = \frac{y_2 – y_1}{x_2 – x_1} = \frac{7 – 3}{5 – 2} = \frac{4}{3} \]
Nzọụkwụ nke 3: Nha nke ahịrị tangent kwụ ọtọ na nha nke radius (nhazi nke ngwaahịa ahụ bụ -1):
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \]
Nzọụkwụ nke 4: Chọpụta nha nhata nke ahịrị tangent site na iji isi ihe \( P (5, 7) \):
\[ y – y_1 = m (x – x_1) \]
\[ y – 7 = -\frac{3}{4} (x – 5) \]
Mee ka ọ dị mfe:
\[ y – 7 = -\frac{3}{4}x + \frac{15}{4} \]
\[ afọ 4 – 28 = -3x + 15 \]
\[ 3x + 4y – 43 = 0 \]
Ya mere, usoro nke ahịrị tangent bụ:
\[ 3x + 4y – 43 = 0 \]
Ajụjụ nke 2: Ịchọpụta Isi Ihe Dị Iche na Nha Ahịrị
Ajụjụ:
E nyere okirikiri nwere nha anya \( x^2 + y^2 = 25 \) na ahịrị \( y = \frac{3}{4}x + 2 \). Chọpụta ebe tangency dị n'etiti ahịrị na okirikiri ahụ.
Azịza:
Nzọụkwụ 1: Tinye nha nha ahịrị ahụ n'ime nha nha nke okirikiri ahụ:
Nha nhata nke okirikiri:
\[ x^2 + y^2 = 25 \]
Tinye \( y = \frac{3}{4}x + 2 \) n'ime nha okirikiri:
\[ x^2 + \left(\frac{3}{4}x + 2\nri)^2 = 25 \]
\[ x^2 + \left(\frac{9}{16}x^2 + \frac{12}{4}x + 4 \nri) = 25 \]
\[ x^2 + \frac{9}{16}x^2 + \frac{6}{2}x + 4 = 25 \]
\[ x^2 + \frac{9}{16}x^2 + 3x + 4 = 25 \]
Nzọụkwụ nke 2: Mee ka usoro ahụ dị mfe:
\[ 16x^2 + 9x^2 + 48x + 64 = 400 \]
\[ 25x^2 + 48x + 64 – 400 = 0 \]
\[ 25x^2 + 48x – 336 = 0 \]
Nzọụkwụ nke 3: Ịchọta mgbọrọgwụ site na iji usoro quadratic:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
\[ a = 25, b = 48, c = -336 \]
\[ x = \frac{-48 \pm \sqrt{48^2 – 4 \cdot 25 \cdot (-336)}}{2 \cdot 25} \]
\[ x = \frac{-48 \pm \sqrt{2304 + 33600}}{50} \]
\[ x = \frac{-48 \pm \sqrt{35904}}{50} \]
\[ x = \frac{-48 \pm 189.501}{50} \]
Ịhọrọ \( x \) dị irè dabere na isi ihe na-akpata nrụgide (naanị otu \( x \) ga-emepụta isi ihe na-akpata nrụgide):
\[ x = \frac{141.501}{50} \ihe dị ka 2.83 \]
\[ x \ihe dị ka 2.83 \]
Nzọụkwụ nke 4: Tinye \( x \) n'ime nha nha ahịrị iji nweta \( y \):
\[ y = \frac{3}{4}(2.83) + 2 \]
\[ y \ihe dịka 2.12 + 2 \]
\[ y \ihe dịka 4.12 \]
Ya mere, isi ihe dị n'etiti ahịrị \( y = \frac{3}{4}x + 2 \) na okirikiri \( x^2 + y^2 = 25 \) bụ \( (2.83, 4.12) \).
Mmechi
Ịmụta echiche nke okirikiri na tangent gụnyere ịghọta isi ihe dị na geometry na ikike idozi nsogbu site na iji usoro mgbakọ na mwepụ. Nsogbu ndị dị ka ndị dị n'elu na-enyere ụmụ akwụkwọ aka ịmụta itinye ozizi n'ọrụ n'ọnọdụ ndị doro anya. Site na omume na-aga n'ihu, a na-atụ anya ka ụmụ akwụkwọ ghọta ma dozie nsogbu ngwa ngwa.
