Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o nā Tangents i nā ʻāpana Conic
Pendahuluan
ʻO ka ʻāpana conic kahi piʻo i loaʻa mai ka hui ʻana o kahi mokulele me kahi cone dipolar. ʻO kēia mau piʻo e komo pū me nā pōʻai, ellipses, parabolas, a me nā hyperbolas. ʻO kahi kumuhana koʻikoʻi i ka hoʻomaopopo ʻana i nā ʻāpana conic ʻo ia ka laina tangent. ʻO ka tangent i kahi ʻāpana conic he laina e hoʻopā ana i ka piʻo conic ma hoʻokahi wale nō kiko. E kūkākūkā kēia ʻatikala i kekahi mau pilikia hoʻohālike a me kahi kūkākūkā ʻana o nā tangents i nā ʻāpana conic.
ʻO ke kihi i kahi pōʻai
He ʻāpana conic ka pōʻai nona ke ʻano maʻalahi loa a me ka symmetry kūpono. E hoʻomaka kākou me kahi pilikia hoʻohālike e pili ana i nā tangents i kahi pōʻai.
Laʻana Nīnau 1
Hāʻawi ʻia kahi pōʻai me ka hoohalike \( (x – 2)^2 + (y + 3)^2 = 25 \). E hoʻoholo i ka hoohalike o ka laina tangent ma ke kiko \((5, -3)\) ma ka pōʻai.
Pahana
ʻO ke kaulike laulā o kahi pōʻai ʻo \( (x – h)^2 + (y – k)^2 = r^2 \), me \( (h, k) \) ma ke kikowaena o ka pōʻai a me \( r \) ma ke radius. Ma kēia pilikia, ʻo ke kikowaena o ka pōʻai \((h, k)\) ʻo \((2, -3)\) a ʻo ka radius \( r = \sqrt{25} = 5 \).
Hiki ke loaʻa ka laina tangent ma ke kiko \((x_1, y_1)\) ma ka pōʻai me ka hoʻohana ʻana i kēia haʻilula:
\[ (x – h)(x_1 – h) + (y – k)(y_1 – k) = r^2 \]
E hoʻokomo i nā waiwai i ʻike ʻia:
\[ (x – 2)(5 – 2) + (y + 3)(-3 + 3) = 25 \]
\[ (x – 2)(3) + (y + 3)(0) = 25 \]
\[ 3(x – 2) = 25 \]
\[ 3x – 6 = 25 \]
\[ 3x = 31 \]
\[ x = \frac{31}{3} \]
ʻO ke kaulike o ka laina tangent ʻo \(x = \frac{31}{3}\), akā aia kahi hewa ma kēia ʻano hana no ka mea ʻo ke kiko \((5, -3)\) he kiko maopopo ia ma ka pōʻai. No laila, hoʻohana mākou i ke ʻano hana kuʻuna ma ke pani ʻana i ka pali o ka laina tangent ma kēia kiko kikoʻī:
ʻO ke kiko tangent, \((5, -3)\), a laila, ʻo ke gradient (m) o ka laina radius ʻo \(m = \frac{-3 – (-3)}{5 – 2}=0\), kahi i lilo ai ke gradient o ka laina tangent i mea wehewehe ʻole no ka tangent kū pololei ma mua.
Ka Laina Tangent i kahi Ellipse
He ʻāpana conic ka ellipse me ʻelua mau axes o symmetry: kahi axis nui (lōʻihi) a me kahi axis liʻiliʻi (pōkole). Eia kekahi mau laʻana o nā pilikia me nā ellipses.
Laʻana Nīnau 2
Hāʻawi ʻia kahi ellipse me ka hoohalike \(\frac{x^2}{16} + \frac{y^2}{9} = 1\). E hoʻoholo i ka hoohalike o ka laina tangent ma ke kiko \((2, \frac{3}{2})\) ma ka ellipse.
Pahana
ʻO ke kaulike o ka laina tangent i ka ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) ma ke kiko \((x_1, y_1)\) penei:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Me \(a = 4\) a me \(b = 3\), e pani i nā waiwai o \(a\), \(b\), a me ke kiko \((2, \frac{3}{2})\):
\[ \frac{x(2)}{4^2} + \frac{y(\frac{3}{2})}{3^2} = 1 \]
\[ \frac{2x}{16} + \frac{3y}{6} = 1 \]
\[ \frac{x}{8} + \frac{y}{2} = 1 \]
E hoʻonui i ka hoohalike holoʻokoʻa me 8 e hoʻopau i nā hakina:
\[ x + 4y = 8 \]
No laila, ʻo ke kaulike o ka laina tangent i ka ellipse ʻo \( x + 4y = 8 \).
Ka Laina Paʻa i ka Parabola
He ʻāpana conic ka parabola me hoʻokahi axis o ka symmetry a me hoʻokahi vertex. Eia kekahi mau laʻana o nā pilikia me nā parabolas.
Laʻana Nīnau 3
Hāʻawi ʻia kahi parabola me ka hoohalike \( y^2 = 4x \). E hoʻoholo i ka hoohalike o ka laina tangent ma ke kiko \((1, 2)\) ma ka parabola.
Pahana
ʻO ke kaulike o ka laina tangent i ka parabola \( y^2 = 4ax \) ma ke kiko \((x_1, y_1)\) penei:
\[ yy_1 = 2a(x + x_1) \]
Mai ke kaulike parabola \( y^2 = 4x \), loaʻa iā mākou \( 4a = 4 \) i hiki ai \( a = 1 \). E pani i ka waiwai o \( a \) a me ke kiko \((1, 2)\):
2y = 2(1)(x + 1)
\[ 2y = 2x + 2 \]
\[ y = x + 1 \]
No laila, ʻo ke kaulike o ka laina tangent i ka parabola ʻo \( y = x + 1 \).
Ka Laina Paʻa i ka Hyperbola
He ʻāpana conic ka hyperbola me ʻelua lālā a me ʻelua asymptotes. Eia kekahi mau laʻana o nā pilikia me nā hyperbolas.
Laʻana Nīnau 4
Hāʻawi ʻia kahi hyperbola me ka hoohalike \( \frac{x^2}{25} – \frac{y^2}{16} = 1 \). E hoʻoholo i ka hoohalike o ka laina tangent ma ke kiko \((5, 0)\) ma ka hyperbola.
Pahana
ʻO ke kaulike o ka laina tangent i ka hyperbola \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\) ma ke kiko \((x_1, y_1)\) penei:
\[ \frac{xx_1}{a^2} – \frac{yy_1}{b^2} = 1 \]
Me \( a = 5 \) a me \( b = 4 \), e pani i nā waiwai \( a \), \( b \), a me ke kiko \((5, 0)\):
\[ \frac{x(5)}{25} – \frac{y(0)}{16} = 1 \]
\[ \frac{5x}{25} – 0 = 1 \]
\[ \frac{x}{5} = 1 \]
\[ x = 5 \]
No laila, ʻo ke kaulike o ka laina tangent i ka hyperbola ʻo \( x = 5 \).
Ka hopena
He kuleana koʻikoʻi ko nā tangents i nā ʻāpana conic i ka makemakika a me nā noi hana like ʻole. ʻO ka hoʻomaopopo ʻana i ke ʻano o ka loaʻa ʻana o nā kaulike o nā tangents i nā ʻano ʻāpana conic like ʻole, e like me nā pōʻai, ellipses, parabolas, a me hyperbolas, he mākaukau koʻikoʻi ia i ka calculus a me ke geometry analytic. Me nā laʻana a me nā kūkākūkā ma luna, ua manaʻolana ʻia e loaʻa i ka poʻe heluhelu kahi ʻike maikaʻi aʻe o nā manaʻo a me nā ʻano hana no ka hoʻoholo ʻana i nā tangents i nā ʻāpana conic.
