Mafunso a Zitsanzo ndi Kukambirana za Ma Tangents ku Magawo a Conic
Pendauluan
Gawo la conic ndi curve yomwe imachokera ku malo olumikizirana a ndege ndi dipolar cone. Ma curve amenewa akuphatikizapo ma circles, ellipses, parabolas, ndi hyperbolas. Mutu umodzi wofunikira pakumvetsetsa magawo a conic ndi mzere wa tangent. Tangent ku gawo la conic ndi mzere womwe umakhudza curve ya conic pamalo amodzi okha. Nkhaniyi ikambirana zitsanzo zingapo za mavuto ndi kukambirana za tangents ku magawo a conic.
Kuzungulira kwa Mzere
Bwalo ndi gawo la conic lomwe lili ndi mawonekedwe osavuta komanso ofanana bwino. Tiyeni tiyambe ndi chitsanzo cha vuto lokhudza ma tangents ku bwalo.
Chitsanzo cha Funso 1
Yapatsidwa bwalo lokhala ndi equation \( (x – 2)^2 + (y + 3)^2 = 25 \). Dziwani equation ya mzere wa tangent pamalo \((5, -3)\) pa bwalo.
Zokambirana
Chiyerekezo cha bwalo ndi \( (x – h)^2 + (y – k)^2 = r^2 \), ndi \( (h, k) \) ngati pakati pa bwalo ndi \( r \) ngati radius. Mu vutoli, pakati pa bwalo \((h, k)\) ndi \((2, -3)\) ndipo radius \( r = \sqrt{25} = 5 \).
Mzere wozungulira womwe uli pamalo \((x_1, y_1)\) pa bwalo ungapezeke pogwiritsa ntchito njira iyi:
\[ (x – h)(x_1 – h) + (y – k)(y_1 – k) = r^2 \]
Lowetsani mfundo zodziwika bwino:
\[ (x – 2) (5 – 2) + (y + 3)(-3 + 3) = 25 \]
\[ (x – 2)(3) + (y + 3)(0) = 25 \]
\[ 3(x – 2) = 25 \]
\[ 3x – 6 = 25 \]
\[ 3x = 31 \]
\[ x = \frac{31}{3} \]
Equation ya mzere wa tangent ndi \(x = \frac{31}{3}\), koma pali cholakwika mu njira iyi chifukwa mfundo \((5, -3)\) ndi mfundo yowonekera bwino pa bwalo. Chifukwa chake, timagwiritsa ntchito njira yachikhalidwe posintha malo otsetsereka a mzere wa tangent pamalo awa enieni:
Mfundo yozungulira ndi, \((5, -3)\), ndiye kuti, gradient (m) ya mzere wozungulira ndi \(m = \frac{-3 – (-3)}{5 – 2}=0\), pomwe gradient ya mzere wozungulira imakhala yosafotokozedwa bwino ya vertical tangent yapitayi.
Mzere Wolunjika ku Ellipse
Duwa lozungulira ndi gawo la conic lomwe lili ndi ma axes awiri ofanana: axis yayikulu (yaitali) ndi axis yaying'ono (yaifupi). Nazi zitsanzo za mavuto ndi ma ellipses.
Chitsanzo cha Funso 2
Kupatsidwa ellipse yokhala ndi equation \(\frac{x^2}{16} + \frac{y^2}{9} = 1\). Dziwani equation ya mzere wa tangent pamalo \((2, \frac{3}{2})\) pa ellipse.
Zokambirana
Chiyerekezo cha mzere wa tangent kupita ku ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) pamalo \((x_1, y_1)\) ndi:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Ndi \(a = 4\) ndi \(b = 3\), m'malo mwa ma values a \(a\), \(b\), ndi mfundo \((2, \frac{3}{2})\):
\[ \frac{x(2)}{4^2} + \frac{y(\frac{3}{2})}{3^2} = 1 \]
\[ \frac{2x}{16} + \frac{3y}{6} = 1 \]
\[ \frac{x}{8} + \frac{y}{2} = 1 \]
Chulukitsani equation yonse ndi 8 kuti muchotse magawo:
\[x + 4y = 8 \]
Kotero, equation ya mzere wa tangent ku ellipse ndi \( x + 4y = 8 \).
Mzere wa Tangent kupita ku Parabola
Parabola ndi gawo la conic lomwe lili ndi mzere umodzi wofanana ndi vertex imodzi. Nazi zitsanzo za mavuto omwe ali ndi parabola.
Chitsanzo cha Funso 3
Tapatsidwa parabola yokhala ndi equation \( y^2 = 4x \). Dziwani equation ya mzere wa tangent pamalo \((1, 2)\) pa parabola.
Zokambirana
Chiyerekezo cha mzere wa tangent ku parabola \( y^2 = 4ax \) pamalo \((x_1, y_1)\) ndi:
\[ yy_1 = 2a(x + x_1) \]
Kuchokera ku parabola equation \( y^2 = 4x \), timapeza \( 4a = 4 \) kotero kuti \( a = 1 \). M'malo mwa mtengo wa \( a \) ndi mfundo \((1, 2)\):
\[ 2y = 2(1)(x + 1) \]
\[ 2y = 2x + 2 \]
\[ y = x + 1 \]
Kotero, equation ya mzere wa tangent ku parabola ndi \( y = x + 1 \).
Mzere wa Tangent kupita ku Hyperbola
Hyperbola ndi gawo la conic lomwe lili ndi nthambi ziwiri ndi ma asymptotes awiri. Nazi zitsanzo za mavuto a hyperbolas.
Chitsanzo cha Funso 4
Kupatsidwa hyperbola yokhala ndi equation \( \frac{x^2}{25} – \frac{y^2}{16} = 1 \). Dziwani equation ya mzere wa tangent pamalo \((5, 0)\) pa hyperbola.
Zokambirana
Chiyerekezo cha mzere wa tangent ku hyperbola \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\) pamalo \((x_1, y_1)\) ndi:
\[ \frac{xx_1}{a^2} – \frac{yy_1}{b^2} = 1 \]
Ndi \( a = 5 \) ndi \( b = 4 \), m'malo mwa ma values \( a \), \( b \), ndi mfundo \((5, 0)\):
\[ \frac{x(5)}{25} – \frac{y(0)}{16} = 1 \]
\[ \frac{5x}{25} – 0 = 1 \]
\[ \frac{x}{5} = 1 \]
\[x = 5 \]
Kotero, equation ya mzere wa tangent ku hyperbola ndi \( x = 5 \).
Mapeto
Ma tangenti ku magawo a conic amachita gawo lofunika kwambiri pa masamu ndi ntchito zosiyanasiyana zothandiza. Kumvetsetsa momwe mungapezere ma equation a ma tangenti ku mitundu yosiyanasiyana ya magawo a conic, monga ma circles, ellipses, parabolas, ndi hyperbolas, ndi luso lofunikira pa calculus ndi analytic geometry. Ndi zitsanzo ndi zokambirana zomwe zili pamwambapa, tikukhulupirira kuti owerenga adzamvetsetsa bwino malingaliro ndi njira zodziwira ma tangenti ku magawo a conic.
