Mibvunzo yemuenzaniso nekukurukurirana kwezvikamu zveTangents kuConic
Pendauluan
Chikamu checonic ikombama inobva pakusangana kweplane ne dipolar cone. Makombama aya anosanganisira madenderedzwa, ma ellipses, ma parabolas, uye ma hyperbolas. Imwe nyaya inokosha pakunzwisisa zvikamu zveconic mutsetse we tangent. Chikamu che tangent kuchikamu checonic mutsetse unobata kona yeconic panzvimbo imwe chete. Chinyorwa chino chichakurukura mienzaniso yakati wandei yezvinetso uye hurukuro ye tangents kuzvikamu zveconic.
Kubatana neDenderedzwa
Denderedzwa chikamu checonic chine chimiro chiri nyore uye chakaringana zvakakwana. Ngatitangei nemuenzaniso wedambudziko pamusoro pematanho edenderedzwa.
Muenzaniso Mubvunzo 1
Kupiwa denderedzwa rine equation \( (x – 2)^2 + (y + 3)^2 = 25 \). Sarudza equation yemutsetse wetangent panzvimbo \((5, -3)\) padenderedzwa.
Kukurukurirana
Muezaniso wedenderedzwa unonzi \( (x – h)^2 + (y – k)^2 = r^2 \), ne \( (h, k) \) sepakati pedenderedzwa uye \( r \) seredhiyo. Mudambudziko iri, pakati pedenderedzwa \((h, k)\) ndi \((2, -3)\) uye redhiyo \( r = \sqrt{25} = 5 \).
Mutsetse we tangent panzvimbo \((x_1, y_1)\) padenderedzwa unogona kuwanikwa uchishandisa fomura inotevera:
\[ (x – h)(x_1 – h) + (y – k)(y_1 – k) = r^2 \]
Isa kukosha kunozivikanwa:
\[ (x – 2)(5 – 2) + (y + 3)(-3 + 3) = 25 \]
\[ (x – 2)(3) + (y + 3)(0) = 25 \]
\[ 3(x – 2) = 25 \]
\[3x – 6 = 25 \]
\[ 3x = 31 \]
\[ x = \frac{31}{3} \]
Equation yemutsetse wetangent ndeye \(x = \frac{31}{3}\), asi pane chikanganiso munzira iyi nekuti poindi \((5, -3)\) zviri pachena kuti poindi iri padenderedzwa. Saka, tinoshandisa nzira yechinyakare nekutsiva mutsetse wetangent panzvimbo iyi chaiyo:
Poindi yetangent ndeye, \((5, -3)\), saka, gradient (m) yemutsetse weradius ndiyo \(m = \frac{-3 – (-3)}{5 – 2}=0\), apo gradient yemutsetse wetangent inova isina kutsanangurwa kune tangent yakatarisana.
Mutsetse weTangent kuenda kuEllipse
Duramazwi rakaita sedenderedzwa (ellipse) chikamu checonic chine mativi maviri ekuenzana: denderedzwa guru (refu) uye denderedzwa diki (pfupi). Heano mimwe mienzaniso yematambudziko ane ellipse.
Muenzaniso Mubvunzo 2
Kupiwa ellipse ine equation \(\frac{x^2}{16} + \frac{y^2}{9} = 1\). Sarudza equation yemutsetse wetangent panzvimbo \((2, \frac{3}{2})\) paellipse.
Kukurukurirana
Equation yemutsetse wetangent kuenda ku ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) panzvimbo \((x_1, y_1)\) ndeiyi:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Ne \(a = 4\) uye \(b = 3\), tsivai kukosha kwe \(a\), \(b\), uye poindi \((2, \frac{3}{2})\):
\[ \frac{x(2)}{4^2} + \frac{y(\frac{3}{2})}{3^2} = 1 \]
\[ \frac{2x}{16} + \frac{3y}{6} = 1 \]
\[ \frac{x}{8} + \frac{y}{2} = 1 \]
Wedzera equation yese ne8 kuti ubvise zvikamu:
\[x + 4y = 8 \]
Saka, equation yemutsetse wetangent kune ellipse ndeye \( x + 4y = 8 \).
Mutsetse weTangent kuenda kuParabola
Parabola chikamu checonic chine axis imwe chete yekuenzanisa uye vertex imwe chete. Heano mimwe mienzaniso yematambudziko ane parabolas.
Muenzaniso Mubvunzo 3
Kupiwa parabola ine equation \( y^2 = 4x \). Sarudza equation yemutsetse wetangent panzvimbo \((1, 2)\) paparabola.
Kukurukurirana
Equation yemutsetse wetangent kune parabola \( y^2 = 4ax \) panzvimbo \((x_1, y_1)\) ndeiyi:
\[ yy_1 = 2a(x + x_1) \]
Kubva mu parabola equation \( y^2 = 4x \), tinowana \( 4a = 4 \) kuitira kuti \( a = 1 \). Tsiva kukosha kwe \( a \) uye poindi \((1, 2)\):
\[ 2y = 2(1)(x + 1) \]
\[ 2y = 2x + 2 \]
\[ y = x + 1 \]
Saka, equation yemutsetse wetangent kune parabola ndi \( y = x + 1 \).
Mutsetse weTangent kuenda kuHyperbola
Hyperbola chikamu checonic chine matavi maviri uye maviri asymptotes. Heano mimwe mienzaniso yematambudziko ane hyperbolas.
Muenzaniso Mubvunzo 4
Kupiwa hyperbola ine equation \( \frac{x^2}{25} – \frac{y^2}{16} = 1 \). Sarudza equation yemutsetse wetangent panzvimbo \((5, 0)\) pahyperbola.
Kukurukurirana
Equation yemutsetse wetangent kune hyperbola \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\) panzvimbo \((x_1, y_1)\) ndeiyi:
\[ \frac{xx_1}{a^2} – \frac{yy_1}{b^2} = 1 \]
Na \( a = 5 \) uye \( b = 4 \), tsivai kukosha \( a \), \( b \), uye poindi \((5, 0)\):
\[ \frac{x(5)}{25} – \frac{y(0)}{16} = 1 \]
\[ \frac{5x}{25} – 0 = 1 \]
\[ \frac{x}{5} = 1 \]
\[x = 5 \]
Saka, equation yemutsetse wetangent kune hyperbola ndeye \( x = 5 \).
Mhedziso
Matauriro ezvikamu zveconic anoita basa rakakosha mumasvomhu uye mashandisirwo akasiyana-siyana anoshanda. Kunzwisisa mawaniro ekuenzanisa matauriro ezvikamu zvakasiyana-siyana zveconic, zvakaita sedenderedzwa, maellipses, maparabolas, uye mahyperbolas, hunyanzvi hwakakosha mukuverenga uye kuongorora geometry. Nemienzaniso nehurukuro dziri pamusoro apa, zvinotarisirwa kuti vaverengi vachanzwisisa zviri nani pfungwa nenzira dzekusarudza matauriro ezvikamu zveconic.
