Lipotso tsa Mehlala le Puisano ea Li-Circle le Li-Tangents
Didikadikwe le di-tangent ke dihlooho tse pedi tse tshohlwang kgafetsa dipalong, haholo-holo boemong ba sekolo se mahareng. Ho utlwisisa mohopolo le tshebediso ya di-tangent didikadikweng ho bohlokwa bakeng sa ho tebisa tsebo ya hao ya jeometri. Sengolwa sena se tla fana ka mehlala ya mathata le dipuisano ka didikadikwe le di-tangent ho fa babali kutlwisiso e tebileng.
Selelekela Khopolo-taba ea Li-circles le Tangents
Selikalikoe
Selikalikoe ke sete ea lintlha tse ka har'a sefofane tse hole ka ho lekana le ntlha e tsitsitseng e bitsoang bohareng ba selikalikoe. Sebaka sena se tsitsitseng se tsejoa e le radius ea selikalikoe. Ka lipalo, selikalikoe se ka hlalosoa ka equation:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
moo \((a, b)\) e leng dikhokahano tsa bohareng ba sedikadikwe mme \(r\) e le radius.
Tangent
Tangent ho selikalikoe ke mola o amang selikalikoe ntlheng e le 'ngoe hantle. Ntlha ena e bitsoa ntlha ea tangency. Tšobotsi e ka sehloohong ea tangent ke hore e otlolohile ho radius e huloang ho tloha bohareng ba selikalikoe ho ea ntlheng ea tangency.
Lipotso tsa Mehlala le Puisano
Potso ea 1: Ho Fumana Tekanyo ea Mola oa Tangent
Potso:
Fanoa ka selikalikoe se nang le setsi ho \( (2, 3) \) le radius 5. Fumana equation ea mola o tangent ho selikalikoe ntlheng \( P \) ka li-coordinates \( (5, 7) \).
Puisano:
Mohato oa 1: Etsa bonnete ba hore ntlha \( P \) ehlile e selikalikoe.
Ho hlahloba hore na \( P (5, 7) \) e hodima selikalikoe se nang le setsi \( (2, 3) \) le radius \( 5 \), kenya dikhokahano tsa \( P \) sebakeng sa equation ya selikalikoe:
\[ (x – 2)^2 + (y – 3)^2 = 5^2 \]
\[ (5 – 2)^2 + (7 – 3)^2 = 25 \]
\[ 3^2 + 4^2 = 25 \]
\[9 + 16 = 25 \]
Kaha tekano ke 'nete, ntlha \( P \) e lutse selikalikoeng.
Mohato oa 2: Fumana hore na radius e feta joang ho \( (2, 3) \) le \( (5, 7) \):
\[ m_{radius} = \frac{y_2 – y_1}{x_2 – x_1} = \frac{7 – 3}{5 – 2} = \frac{4}{3} \]
Mohato oa 3: Ho phahama ha mola o otlolohileng ho ea ho ho phahama ha radius (ho phahama ha sehlahisoa ke -1):
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \]
Mohato oa 4: Fumana equation ea mola o nang le tangent u sebelisa ntlha \( P (5, 7) \):
\[ y – y_1 = m (x – x_1) \]
\[ y – 7 = -\frac{3}{4} (x – 5) \]
Nolofatsa:
\[ y – 7 = -\frac{3}{4}x + \frac{15}{4} \]
\[ 4y – 28 = -3x + 15 \]
\[ 3x + 4y – 43 = 0 \]
Kahoo, equation ea mola oa tangent ke:
\[ 3x + 4y – 43 = 0 \]
Potso ea 2: Ho Fumana Ntlha ea Tangency ho tsoa ho Line Equation
Potso:
Fanoa ka selikalikoe se nang le equation \( x^2 + y^2 = 25 \) le mola \( y = \frac{3}{4}x + 2 \). Fumana ntlha ea ho kopana pakeng tsa mola le selikalikoe.
Puisano:
Mohato oa 1: Kenya equation ea mola sebakeng sa equation ea selikalikoe:
Tekanyo ea selikalikoe:
\[ x^2 + y^2 = 25 \]
Kenya \( y = \frac{3}{4}x + 2 \) sebakeng sa equation ya sedikadikwe:
\[ x^2 + \left(\frac{3}{4}x + 2\right)^2 = 25 \]
\[ x^2 + \left(\frac{9}{16}x^2 + \frac{12}{4}x + 4 \right) = 25 \]
\[ x^2 + \frac{9}{16}x^2 + \frac{6}{2}x + 4 = 25 \]
\[ x^2 + \frac{9}{16}x^2 + 3x + 4 = 25 \]
Mohato oa 2: Nolofatsa equation:
\[ 16x^2 + 9x^2 + 48x + 64 = 400 \]
\[ 25x^2 + 48x + 64 – 400 = 0 \]
\[ 25x^2 + 48x – 336 = 0 \]
Mohato oa 3: Ho fumana metso ho sebelisoa foromo ea quadratic:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
\[ a = 25, b = 48, c = -336 \]
\[ x = \frac{-48 \pm \sqrt{48^2 – 4 \cdot 25 \cdot (-336)}}{2 \cdot 25} \]
\[ x = \frac{-48 \pm \sqrt{2304 + 33600}}{50} \]
\[ x = \frac{-48 \pm \sqrt{35904}}{50} \]
\[ x = \frac{-48 \pm 189.501}{50} \]
Ho kgetha \( x \) e nepahetseng ho latela ntlha ya tangency (e le nngwe feela \( x \) e tla hlahisa ntlha ya tangency):
\[ x = \frac{141.501}{50} \hoo e ka bang 2.83 \]
\[x \hoo e ka bang 2.83 \]
Mohato oa 4: Kenya \( x \) sebakeng sa equation ea mola ho fumana \( y \):
\[ y = \frac{3}{4}(2.83) + 2 \]
\[ y \hoo e ka bang 2.12 + 2 \]
\[ y \hoo e ka bang 4.12 \]
Kahoo, ntlha ea ho rarahana pakeng tsa mola \( y = \frac{3}{4}x + 2 \) le selikalikoe \( x^2 + y^2 = 25 \) ke \( (2.83, 4.12) \).
Qetello
Ho tseba mehopolo ea li-circles le li-tangent ho kenyelletsa ho utloisisa metheo ea jeometri le bokhoni ba ho rarolla mathata ka ho sebelisa li-equation tsa lipalo. Mathata a kang a kaholimo a thusa baithuti ho itloaetsa ho sebelisa khopolo-taba maemong a hlakileng haholoanyane. Ka ho itloaetsa ho tsitsitseng, baithuti ba lebelletsoe ho utloisisa le ho rarolla mathata habonolo.
