Cov Lus Nug Piv Txwv thiab Kev Sib Tham Txog Tangents rau Conic Sections
Pendahuluuan
Ib ntu conic yog ib txoj kab nkhaus uas tshwm sim los ntawm kev sib tshuam ntawm lub dav hlau nrog lub dipolar cone. Cov kab nkhaus no suav nrog lub voj voog, ellipses, parabolas, thiab hyperbolas. Ib qho tseem ceeb hauv kev nkag siab txog cov ntu conic yog kab tangent. Ib txoj kab tangent rau ib ntu conic yog ib txoj kab uas kov cov kab conic ntawm ib qho chaw xwb. Tsab xov xwm no yuav tham txog ntau qhov teeb meem piv txwv thiab kev sib tham txog tangents rau cov ntu conic.
Tangent rau ib lub voj voog
Ib lub voj voog yog ib ntu conic uas muaj cov duab yooj yim tshaj plaws thiab muaj qhov sib npaug zoo meej. Cia peb pib nrog ib qho piv txwv ntawm qhov teeb meem txog kev sib tshuam rau lub voj voog.
Piv txwv lus nug 1
Muab ib lub voj voog uas muaj tus qauv \( (x – 2)^2 + (y + 3)^2 = 25 \). Txheeb xyuas tus qauv ntawm txoj kab tangent ntawm qhov chaw \((5, -3)\) ntawm lub voj voog.
Kev Sib Tham
Tus qauv dav dav ntawm lub voj voog yog \( (x – h)^2 + (y – k)^2 = r^2 \), nrog \( (h, k) \) ua qhov chaw nruab nrab ntawm lub voj voog thiab \( r \) ua lub vojvoog. Hauv qhov teeb meem no, qhov chaw nruab nrab ntawm lub voj voog \((h, k)\) yog \((2, -3)\) thiab lub vojvoog \( r = \sqrt{25} = 5 \).
Cov kab tangent ntawm qhov point \((x_1, y_1)\) ntawm lub voj voog tuaj yeem pom siv cov qauv hauv qab no:
\[ (x – h)(x_1 – h) + (y – k)(y_1 – k) = r^2 \]
Sau cov nqi uas paub lawm:
\[ (x – 2)(5 – 2) + (y + 3)(-3 + 3) = 25 \]
\[ (x – 2)(3) + (y+3)(0) = 25 \]
\[ 3(x – 2) = 25 \]
\[ 3x – 6 = 25 \]
\[ 3x = 31 \]
\[ x = \frac{ 31}{ 3} \]
Tus qauv ntawm txoj kab tangent yog \(x = \frac{31}{3}\), tab sis muaj qhov yuam kev hauv txoj kev no vim tias qhov taw tes \((5, -3)\) yog qhov taw tes ntawm lub voj voog. Yog li ntawd, peb siv txoj kev ib txwm los ntawm kev hloov qhov nqes hav ntawm txoj kab tangent ntawm qhov taw tes tshwj xeeb no:
Lub tangent point yog, \((5, -3)\), ces, qhov gradient (m) ntawm txoj kab radius yog \(m = \frac{-3 – (-3)}{5 – 2}=0\), qhov twg qhov gradient ntawm txoj kab tangent tsis tau txhais rau qhov tangent ntsug ua ntej.
Kab tangent rau ib qho ellipse
Ib lub ellipse yog ib ntu conic uas muaj ob txoj kab sib luag: ib txoj kab loj (ntev) thiab ib txoj kab me (luv). Nov yog qee qhov piv txwv ntawm cov teeb meem nrog ellipses.
Piv txwv lus nug 2
Muab ib lub ellipse nrog qhov sib npaug \(\frac{x^2}{16} + \frac{y^2}{9} = 1\). Txheeb xyuas qhov sib npaug ntawm txoj kab tangent ntawm qhov chaw \((2, \frac{3}{2})\) ntawm lub ellipse.
Kev Sib Tham
Tus qauv ntawm txoj kab tangent rau lub ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) ntawm qhov point \((x_1, y_1)\) yog:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Nrog \(a = 4\) thiab \(b = 3\), hloov cov nqi ntawm \(a\), \(b\), thiab tus taw tes \((2, \frac{3}{2})\):
\[ \frac{x(2)}{4^2} + \frac{y(3}{2})}{3^2} = 1 \]
\[ \frac{2x}{16} + \frac{3y}{6} = 1 \]
\[ \frac{x}{8} + \frac{y}{2} = 1 \]
Muab tag nrho cov lej sib npaug los ntawm 8 kom tshem tawm cov feem:
\[ x + 4y = 8 \]
Yog li, qhov sib npaug ntawm txoj kab tangent rau lub ellipse yog \( x + 4y = 8 \).
Kab tangent rau Parabola
Ib daim duab parabola yog ib daim duab conic uas muaj ib txoj kab sib luag thiab ib lub vertex. Nov yog qee qhov piv txwv ntawm cov teeb meem nrog parabolas.
Piv txwv lus nug 3
Muab ib daim parabola nrog rau qhov sib npaug \( y^2 = 4x \). Txheeb xyuas qhov sib npaug ntawm txoj kab tangent ntawm qhov chaw \((1, 2)\) ntawm daim parabola.
Kev Sib Tham
Tus qauv ntawm txoj kab tangent rau parabola \( y^2 = 4ax \) ntawm qhov point \((x_1, y_1)\) yog:
\[ yy_1 = 2a(x + x_1) \]
Los ntawm kab zauv parabola \( y^2 = 4x \), peb tau txais \( 4a = 4 \) yog li ntawd \( a = 1 \). Hloov tus nqi ntawm \( a \) thiab qhov taw tes \((1, 2)\):
\[ 2y = 2(1)(x + 1) \]
\[ 2y = 2x + 2 \]
\[ y = x + 1 \]
Yog li, qhov sib npaug ntawm txoj kab tangent rau parabola yog \(y = x + 1 \).
Kab Tangent rau Hyperbola
Ib lub hyperbola yog ib ntu conic nrog ob ceg thiab ob lub asymptotes. Nov yog qee qhov piv txwv ntawm cov teeb meem nrog hyperbolas.
Piv txwv lus nug 4
Muab ib qho hyperbola nrog rau qhov sib npaug \( \frac{x^2}{25} – \frac{y^2}{16} = 1 \). Txheeb xyuas qhov sib npaug ntawm txoj kab tangent ntawm qhov chaw \((5, 0)\) ntawm qhov hyperbola.
Kev Sib Tham
Tus qauv ntawm txoj kab tangent rau hyperbola \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\) ntawm qhov point \((x_1, y_1)\) yog:
\[ \frac{xx_1}{a^2} – \frac{yy_1}{b^2} = 1 \]
Nrog \(a = 5\) thiab \(b = 4\), hloov cov nqi \(a\), \(b\), thiab tus taw tes \((5, 0)\):
\[ \frac{x(5)}{25} – \frac{y(0)}{16} = 1 \]
\[ \frac{5x}{25} – 0 = 1 \]
\[ \frac{x}{5} = 1 \]
\[ x = 5 \]
Yog li, qhov sib npaug ntawm txoj kab tangent rau hyperbola yog \(x = 5 \).
Xaus
Kev sib txuas ntawm cov kab sib txuas rau cov ntu conic ua lub luag haujlwm tseem ceeb hauv kev suav lej thiab ntau yam kev siv. Kev nkag siab txog yuav ua li cas nrhiav cov qauv ntawm cov kab sib txuas rau ntau hom ntu conic, xws li lub voj voog, ellipses, parabolas, thiab hyperbolas, yog ib qho kev txawj tseem ceeb hauv kev suav lej thiab kev sib tham geometry. Nrog rau cov piv txwv thiab kev sib tham saum toj no, vam tias cov neeg nyeem yuav nkag siab zoo dua txog cov ntsiab lus thiab cov txheej txheem rau kev txiav txim siab tangents rau cov ntu conic.
