Imibuzo Eyisibonelo Exoxa Ngesikhundla Somugqa Ngokuphathelene Nendilinga
Ku-geometry, indawo yomugqa ohlobene nendilinga ingumqondo oyisisekelo ovame ukuxoxwa ngawo emazingeni ahlukahlukene emfundo. Umugqa ungathatha izikhundla eziningana maqondana nendilinga, okungukuthi njenge-secant, tangent, noma yangaphandle. Ukuqonda lo mqondo akubalulekile nje kuphela ekuxazululeni izinkinga zezibalo kodwa futhi kuthuthukisa ukuqonda kwethu i-geometry uqobo. Lesi sihloko sizohlola ngokucophelela izinkinga ezahlukahlukene zezibonelo futhi sixoxe ngesikhundla somugqa ohlobene nendilinga.
1. Indawo Yomugqa maqondana neSiyingi
Okokuqala, ake sibheke imiqondo eyisisekelo yezinhlobo ezintathu zezikhundla zomugqa maqondana nendilinga:
1. I-Secant: Umugqa ohlangana nendilinga ezindaweni ezimbili.
2. I-Tangent: Umugqa othinta indilinga endaweni eyodwa kuphela.
3. Umugqa Wangaphandle: Umugqa ongathinti nhlobo indilinga.
2. Ithiyori Eyisisekelo kanye namafomula abalulekile
Amanye amafomula abalulekile kanye nemiqondo eyisisekelo okufanele uyikhumbule:
– Ibanga elisuka endaweni ephakathi yendilinga kuya emgqeni \(d\) linganquma indawo yomugqa maqondana nendilinga:
– Uma \(d > r\) (irediyasi yendilinga), khona-ke umugqa uwumugqa wangaphandle.
– Uma \(d = r\), khona-ke umugqa uwumugqa oqondile.
– Uma \(d < r\), khona-ke umugqa uyi-secant. - Isibalo esijwayelekile sendilinga enesikhungo endaweni \((h,k)\) kanye ne-radius \(r\) ngu \((x - h)^2 + (y - k)^2 = r^2\). - Isibalo somugqa ngesimo esijwayelekile ngu \(Ax + By + C = 0\).
3. Imibuzo Yesibonelo Nengxoxo Isibonelo Umbuzo 1: Umbuzo Ocacile: Unikezwe indilinga ene-equation \((x - 2)^2 + (y + 3)^2 = 25\) kanye nomugqa \(4x + 3y - 7 = 0\). Thola indawo yomugqa maqondana nendilinga. Ingxoxo: 1. Thola isikhungo kanye nerediyasi yendilinga: - Isikhungo sendilinga: \((2,-3)\) - Irediyasi yendilinga: \(r = \sqrt{25} = 5\) 2. Thola ibanga ukusuka enkabeni yendilinga kuya emugqeni: - Sebenzisa ifomula yebanga ukusuka ephuzwini kuya emugqeni: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] - Kulesi simo, \(A = 4\), \(B = 3\), kanye \(C = -7\). Iphuzu eliphakathi lingu \((2, -3)\). - Faka esikhundleni: \[ d = \frac{|4(2) + 3(-3) - 7|}{\sqrt{4^2 + 3^2}} = \frac{|8 - 9 - 7|}{\sqrt{16 + 9}} = \frac{|-8|}{5} = \frac{8}{5} = 1.6 \] 3. Qhathanisa ibanga nerediyasi yendilinga: - \(d = 1.6\) kanye \(r = 5\) - Njengoba \(d < r\), umugqa uyi-secant yendilinga.
Isibonelo Umbuzo 2: Umbuzo Womugqa Oqondile: Uma unikezwe i-equation yendilinga \((x + 2)^2 + (y - 1)^2 = 16\) kanye ne-equation yomugqa \(x + y - 3 = 0\). Ingabe umugqa uthinta indilinga? Uma kunjalo, thola iphuzu le-tangency. Ingxoxo: 1. Thola isikhungo kanye nerediyasi yendilinga: - Isikhungo sendilinga: \((-2, 1)\) - Irediyasi yendilinga: \(r = \sqrt{16} = 4\) 2. Thola ibanga ukusuka maphakathi nendilinga kuya emgqeni: - Sebenzisa ifomula yebanga ukusuka ephuzwini kuya emgqeni: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] - Kulesi simo, \(A = 1\), \(B = 1\), kanye \(C = -3\). Iphuzu eliphakathi lingu-\((-2, 1)\). - Ukufaka esikhundleni: \[ d = \frac{|1(-2) + 1(1) - 3|}{\sqrt{1^2 + 1^2}} = \frac{|-2 + 1 - 3|}{\sqrt{2}} = \frac{|-4|}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \] 3. Qhathanisa ibanga nerediyasi yendilinga: - \(d = 2\sqrt{2}\) kanye \(r = 4\) - Njengoba \(d \neq r\), lo mugqa awuthinti indilinga. Ukulungiswa Nezimpikiswano: - Ibanga elitholiwe alikho ngesimo \(r = 4\) ngakho-ke kumele lihlolwe kabusha uma inkinga inombhalo obhalwe phansi noma ibhalwe kabusha uma kungekho ukulungiswa, umphumela uyafana: lo mugqa awuyona i-tangent kodwa uyi-secant. 4. Imibuzo Yokuzijwayeza
Nazi ezinye zemibuzo yokuzijwayeza ongazizama: 1. Isivivinyo 1: Ukuhlukanisa Imigqa Unikezwe indilinga ene-equation \(x^2 + y^2 = 25\) kanye nomugqa one-equation \(3x + 4y - 20 = 0\). Thola indawo yomugqa maqondana nendilinga. 2. Isivivinyo 2: Imigqa Eqondile Indilinga ine-equation \((x - 1)^2 + (y + 2)^2 = 16\). Ingabe umugqa \(2x - y + 3 = 0\) uyawuthinta umbuthano? Thola iphuzu le-tangency uma kunjalo. 3. Isivivinyo 3: Uhlaka Lwendilinga ene-equation \((x - 4)^2 + (y - 3)^2 = 9\). Thola indawo yomugqa \(x + 2y - 14 = 0\) maqondana nendilinga. Ukuphendula le mibuzo ngokulandela izinyathelo okuxoxwe ngazo kuzokusiza ukuthi uqonde kangcono umqondo wendawo yomugqa maqondana nendilinga. Isiphetho: Ukufunda ubudlelwano bemigqa nendilinga kuyisici esibalulekile se-geometry esingasetshenziswa ezimweni ezahlukahlukene zezemfundo nezisebenzayo. Ngokuqonda imithetho eyisisekelo nokusebenzisa amafomula afanele, singanquma kalula ukuthi umugqa uyahlangana, uthinta, noma ulele ngaphandle kwendilinga. Sithemba ukuthi izincazelo ezikulesi sihloko zizokusiza uthuthukise amakhono akho e-geometry futhi zikulungiselele kangcono izinkinga eziyinkimbinkimbi kakhulu. Ukufunda okuhle!
