Kafli 10

Hvirfilpunktar: $\left(\pm 3,0\right);$ Brennipunktar: $\left(\pm \sqrt{34},0\right)$

$\frac{y^2}{4}-\frac{x^2}{16}=1$

$\frac{{\left(y-3\right)}^2}{25}-\frac{{\left(x-1\right)}^2}{144}=1$

hvirfilpunktar: $\left(\pm 12,0\right);$ co-hvirfilpunktar: $\left(0,\pm 9\right);$ brennipunktar: $\left(\pm 15,0\right);$ aðfellur: $y=\pm \frac{3}{4}x;$

miðja: $\left(3,-4\right);$ hvirfilpunktar: $\left(3,-14\right)$ og $\left(3,6\right);$ co-hvirfilpunktar: $\left(-5,-4\right);$ og $\left(11,-4\right);$ brennipunktar: $\left(3,-4-2\sqrt{41}\right)$ og $\left(3,-4+2\sqrt{41}\right);$ aðfellur: $y=\pm \frac{5}{4}\left(x-3\right)-4$

Hliðar turnsins má lýsa með breiðbogajöfnunni. 𝑥 2 4 0 0 − 𝑦 2 3 6 0 0 = 1 eða 𝑥 2 2 0 2 − 𝑦 2 6 0 2 = 1 . $\frac{x^2}{400}-\frac{y^2}{3600}=1\text{or }\frac{x^2}{{20}^2}-\frac{y^2}{{60}^2}=1.$

Brennipunktur: $\left(-4,0\right);$ Stýrilína: $x=4;$ Endapunktar þverbrennistrengs: $\left(-4,\pm 8\right)$

Brennipunktur: $\left(0,2\right);$ Stýrilína: $y=\mathrm{-2};$ Endapunktar þverbrennistrengs: $\left(\pm 4,2\right).$

$x^2=14y.$

Hvirfilpunktur: $\left(8,-1\right);$ Samhverfuás: $y=\mathrm{-1};$ Brennipunktur: $\left(9,-1\right);$ Stýrilína: $x=7;$ Endapunktar þverbrennistrengs: $\left(9,-3\right)$ og $\left(9,1\right).$

Hvirfilpunktur: $\left(-2,3\right);$ Samhverfuás: $x=\mathrm{-2};$ Brennipunktur: $\left(-2,-2\right);$ Stýrilína: $y=8;$ Endapunktar þverbrennistrengs: $\left(-12,-2\right)$ og $\left(8,-2\right).$

$\frac{{x^{\prime }}^2}{4}+\frac{{y^{\prime }}^2}{1}=1$

sporbaugur; $e=\frac{1}{3};x=-2$

$r=\frac{1}{1-\cos \theta }$

$4-8x+3x^2-y^2=0$

Hann er samhverfur um x-ásinn, y-ásinn og upphafspunktinn.

já; $\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$

já; $\frac{x^2}{{\left(\frac{1}{2}\right)}^2}+\frac{y^2}{{\left(\frac{1}{3}\right)}^2}=1$

$\frac{x^2}{2^2}+\frac{y^2}{7^2}=1;$ Endapunktar langáss $\left(0,7\right)$ og $\left(0,-7\right).$ Endapunktar skammáss $\left(2,0\right)$ og $\left(-2,0\right).$ Brennipunktar í $\left(0,3\sqrt{5}\right),\left(0,-3\sqrt{5}\right).$

$\frac{x^2}{{\left(1\right)}^2}+\frac{y^2}{{\left(\frac{1}{3}\right)}^2}=1;$ Endapunktar langáss $\left(1,0\right)$ og $\left(-1,0\right).$ Endapunktar skammáss $\left(0,\frac{1}{3}\right),\left(0,-\frac{1}{3}\right).$ Brennipunktar í $\left(\frac{2\sqrt{2}}{3},0\right),\left(-\frac{2\sqrt{2}}{3},0\right).$

$\frac{{\left(x-2\right)}^2}{7^2}+\frac{{\left(y-4\right)}^2}{5^2}=1;$ Endapunktar langáss $\left(9,4\right),\left(-5,4\right).$ Endapunktar skammáss $\left(2,9\right),\left(2,-1\right).$ Brennipunktar í $\left(2+2\sqrt{6},4\right),\left(2-2\sqrt{6},4\right).$

$\frac{{\left(x+5\right)}^2}{2^2}+\frac{{\left(y-7\right)}^2}{3^2}=1;$ Endapunktar langáss $\left(-5,10\right),\left(-5,4\right).$ Endapunktar skammáss $\left(-3,7\right),\left(-7,7\right).$ Brennipunktar í $\left(-5,7+\sqrt{5}\right),\left(-5,7-\sqrt{5}\right).$

$\frac{{\left(x-1\right)}^2}{3^2}+\frac{{\left(y-4\right)}^2}{2^2}=1;$ Endapunktar langáss $\left(4,4\right),\left(-2,4\right).$ Endapunktar skammáss $\left(1,6\right),\left(1,2\right).$ Brennipunktar í $\left(1+\sqrt{5},4\right),\left(1-\sqrt{5},4\right).$

$\frac{{\left(x-3\right)}^2}{{\left(3\sqrt{2}\right)}^2}+\frac{{\left(y-5\right)}^2}{{\left(\sqrt{2}\right)}^2}=1;$ Endapunktar langáss $\left(3+3\sqrt{2},5\right),\left(3-3\sqrt{2},5\right).$ Endapunktar skammáss $\left(3,5+\sqrt{2}\right),\left(3,5-\sqrt{2}\right).$ Brennipunktar í $\left(7,5\right),\left(-1,5\right).$

$\frac{{\left(x+5\right)}^2}{{\left(5\right)}^2}+\frac{{\left(y-2\right)}^2}{{\left(2\right)}^2}=1;$ Endapunktar langáss $\left(0,2\right),\left(-10,2\right).$ Endapunktar skammáss $\left(-5,4\right),\left(-5,0\right).$ Brennipunktar í $\left(-5+\sqrt{21},2\right),\left(-5-\sqrt{21},2\right).$

$\frac{{\left(x+3\right)}^2}{{\left(5\right)}^2}+\frac{{\left(y+4\right)}^2}{{\left(2\right)}^2}=1;$ Endapunktar langáss $\left(2,-4\right),\left(-8,-4\right).$ Endapunktar skammáss $\left(-3,-2\right),\left(-3,-6\right).$ Brennipunktar í $\left(-3+\sqrt{21},-4\right),\left(-3-\sqrt{21},-4\right).$

Brennipunktar $\left(-3,-1+\sqrt{11}\right),\left(-3,-1-\sqrt{11}\right)$

Brennipunktur $\left(0,0\right)$

Brennipunktar $\left(-10,30\right),\left(-10,-30\right)$

Miðja $\left(0,0\right),$ Hvirfilpunktar $\left(4,0\right),\left(-4,0\right),(0,3),(0,-3),$ Brennipunktar $\left(\sqrt{7},0\right),\left(-\sqrt{7},0\right)$

Miðja $\left(0,0\right),$ Hvirfilpunktar $\left(\frac{1}{9},0\right),\left(-\frac{1}{9},0\right),\left(0,\frac{1}{7}\right),\left(0,-\frac{1}{7}\right),$ Brennipunktar $\left(0,\frac{4\sqrt{2}}{63}\right),\left(0,-\frac{4\sqrt{2}}{63}\right)$

Miðja $\left(-3,3\right),$ Hvirfilpunktar $\left(0,3\right),\left(-6,3\right),\left(-3,0\right),\left(-3,6\right),$ Brennipunktur $\left(-3,3\right)$

Miðja $\left(1,1\right),$ Hvirfilpunktar $\left(5,1\right),\left(-3,1\right),\left(1,3\right),\left(1,-1\right),$ Brennipunktar $\left(1+2\sqrt{3},1\right),\left(1-2\sqrt{3},1\right)$

Miðja $\left(-4,5\right),$ Hvirfilpunktar $\left(-2,5\right),\left(-6,5\right),\left(-4,6\right),\left(-4,4\right),$ Brennipunktar $\left(-4+\sqrt{3},5\right),\left(-4-\sqrt{3},5\right)$

Miðja $\left(-2,1\right),$ Hvirfilpunktar $\left(0,1\right),\left(-4,1\right),\left(-2,5\right),\left(-2,-3\right),$ Brennipunktar $\left(-2,1+2\sqrt{3}\right),\left(-2,1-2\sqrt{3}\right)$

Miðja $\left(-2,-2\right),$ Hvirfilpunktar $\left(0,-2\right),\left(-4,-2\right),\left(-2,0\right),\left(-2,-4\right),$ Brennipunktur $\left(-2,-2\right)$

$\frac{x^2}{25}+\frac{y^2}{29}=1$

$\frac{{\left(x-4\right)}^2}{25}+\frac{{\left(y-2\right)}^2}{1}=1$

$\frac{{\left(x+3\right)}^2}{16}+\frac{{\left(y-4\right)}^2}{4}=1$

$\frac{x^2}{81}+\frac{y^2}{9}=1$

$\frac{{\left(x+2\right)}^2}{4}+\frac{{\left(y-2\right)}^2}{9}=1$

$\text{Area = 12π}\text{square}\text{units}$

$\text{Area = 2}\sqrt{\text{5}}\text{π}$ ferningseiningar.

$\text{Area = 9π}$ ferningseiningar.

$\frac{x^2}{4h^2}+\frac{y^2}{\frac{1}{4}{h}^2}=1$

$\frac{x^2}{400}+\frac{y^2}{144}=1$ . Distance = 17.32 fet

já $\frac{x^2}{6^2}-\frac{y^2}{3^2}=1$

já $\frac{x^2}{4^2}-\frac{y^2}{5^2}=1$

$\frac{x^2}{5^2}-\frac{y^2}{6^2}=1;$ hvirfilpunktar: $\left(5,0\right),\left(-5,0\right);$ brennipunktar: $\left(\sqrt{61},0\right),\left(-\sqrt{61},0\right);$ aðfellur: $y=\frac{6}{5}x,y=-\frac{6}{5}x$

$\frac{y^2}{2^2}-\frac{x^2}{9^2}=1;$ hvirfilpunktar: $\left(0,2\right),\left(0,-2\right);$ brennipunktar: $\left(0,\sqrt{85}\right),\left(0,-\sqrt{85}\right);$ aðfellur: $y=\frac{2}{9}x,y=-\frac{2}{9}x$

$\frac{{\left(x-1\right)}^2}{3^2}-\frac{{\left(y-2\right)}^2}{4^2}=1;$ hvirfilpunktar: $\left(4,2\right),\left(-2,2\right);$ brennipunktar: $\left(6,2\right),\left(-4,2\right);$ aðfellur: $y=\frac{4}{3}\left(x-1\right)+2,y=-\frac{4}{3}\left(x-1\right)+2$

$\frac{{\left(x-2\right)}^2}{7^2}-\frac{{\left(y+7\right)}^2}{7^2}=1;$ hvirfilpunktar: $\left(9,-7\right),\left(-5,-7\right);$ brennipunktar: $\left(2+7\sqrt{2},-7\right),\left(2-7\sqrt{2},-7\right);$ aðfellur: $y=x-9,y=-x-5$

$\frac{{\left(x+3\right)}^2}{3^2}-\frac{{\left(y-3\right)}^2}{3^2}=1;$ hvirfilpunktar: $\left(0,3\right),\left(-6,3\right);$ brennipunktar: $\left(-3+3\sqrt{2},1\right),\left(-3-3\sqrt{2},1\right);$ aðfellur: $y=x+6,y=-x$

$\frac{{\left(y-4\right)}^2}{2^2}-\frac{{\left(x-3\right)}^2}{4^2}=1;$ hvirfilpunktar: $\left(3,6\right),\left(3,2\right);$ brennipunktar: $\left(3,4+2\sqrt{5}\right),\left(3,4-2\sqrt{5}\right);$ aðfellur: $y=\frac{1}{2}\left(x-3\right)+4,y=-\frac{1}{2}\left(x-3\right)+4$

$\frac{{\left(y+5\right)}^2}{7^2}-\frac{{\left(x+1\right)}^2}{{70}^2}=1;$ hvirfilpunktar: $\left(-1,2\right),\left(-1,-12\right);$ brennipunktar: $\left(-1,-5+7\sqrt{101}\right),\left(-1,-5-7\sqrt{101}\right);$ aðfellur: $y=\frac{1}{10}\left(x+1\right)-5,y=-\frac{1}{10}\left(x+1\right)-5$

$\frac{{\left(x+3\right)}^2}{5^2}-\frac{{\left(y-4\right)}^2}{2^2}=1;$ hvirfilpunktar: $\left(2,4\right),\left(-8,4\right);$ brennipunktar: $\left(-3+\sqrt{29},4\right),\left(-3-\sqrt{29},4\right);$ aðfellur: $y=\frac{2}{5}\left(x+3\right)+4,y=-\frac{2}{5}\left(x+3\right)+4$

$y=\frac{2}{5}\left(x-3\right)-4,y=-\frac{2}{5}\left(x-3\right)-4$

$y=\frac{3}{4}\left(x-1\right)+1,y=-\frac{3}{4}\left(x-1\right)+1$

$\frac{x^2}{9}-\frac{y^2}{16}=1$

$\frac{{\left(x-6\right)}^2}{25}-\frac{{\left(y-1\right)}^2}{11}=1$

$\frac{{\left(x-4\right)}^2}{25}-\frac{{\left(y-2\right)}^2}{1}=1$

$\frac{y^2}{16}-\frac{x^2}{25}=1$

$\frac{y^2}{9}-\frac{{\left(x+1\right)}^2}{9}=1$

$\frac{{\left(x+3\right)}^2}{25}-\frac{{\left(y+3\right)}^2}{25}=1$

$y\left(x\right)=3\sqrt{x^2+1},y\left(x\right)=-3\sqrt{x^2+1}$

$y\left(x\right)=1+2\sqrt{x^2+4x+5},y\left(x\right)=1-2\sqrt{x^2+4x+5}$

$\frac{x^2}{25}-\frac{y^2}{25}=1$

$\frac{x^2}{100}-\frac{y^2}{25}=1$

$\frac{x^2}{400}-\frac{y^2}{225}=1$

$4{\left(x-1\right)}^2-y{2}^2=16$

$\frac{{(x-h)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}={\left(x-3\right)}^2-9y^2=4$

já $x^2=4\left(\frac{1}{16}\right)y$

já ${\left(y-3\right)}^2=4(2)\left(x-2\right)$

$y^2=\frac{1}{8}x,V:(0,0);F:\left(\frac{1}{32},0\right);d:x=-\frac{1}{32}$

$x^2=-\frac{1}{4}y,V:\left(0,0\right);F:\left(0,-\frac{1}{16}\right);d:y=\frac{1}{16}$

$y^2=\frac{1}{36}x,V:\left(0,0\right);F:\left(\frac{1}{144},0\right);d:x=-\frac{1}{144}$

${\left(x-1\right)}^2=4\left(y-1\right),V:\left(1,1\right);F:\left(1,2\right);d:y=0$

${\left(y-4\right)}^2=2\left(x+3\right),V:\left(-3,4\right);F:\left(-\frac{5}{2},4\right);d:x=-\frac{7}{2}$

${\left(x+4\right)}^2=24\left(y+1\right),V:\left(-4,-1\right);F:\left(-4,5\right);d:y=\mathrm{-7}$

${\left(y-3\right)}^2=\mathrm{-12}\left(x+1\right),V:\left(-1,3\right);F:\left(-4,3\right);d:x=2$

${\left(x-5\right)}^2=\frac{4}{5}\left(y+3\right),V:\left(5,-3\right);F:\left(5,-\frac{14}{5}\right);d:y=-\frac{16}{5}$

${\left(x-2\right)}^2=\mathrm{-2}\left(y-5\right),V:\left(2,5\right);F:\left(2,\frac{9}{2}\right);d:y=\frac{11}{2}$

${\left(y-1\right)}^2=\frac{4}{3}\left(x-5\right),V:\left(5,1\right);F:\left(\frac{16}{3},1\right);d:x=\frac{14}{3}$

$x^2=\mathrm{-16}y$

${\left(y-2\right)}^2=4\sqrt{2}\left(x-2\right)$

${\left(y+\sqrt{3}\right)}^2=\mathrm{-4}\sqrt{2}\left(x-\sqrt{2}\right)$

$x^2=y$

${\left(y-2\right)}^2=\frac{1}{4}\left(x+2\right)$

${\left(y-\sqrt{3}\right)}^2=4\sqrt{5}\left(x+\sqrt{2}\right)$

$y^2=\mathrm{-8}x$

${\left(y+1\right)}^2=12\left(x+3\right)$

$\left(0,1\right)$

$x^2=\mathrm{-125}\left(y-20\right),$ hæðin er 7.2 fet

Liðurinn $xy$ veldur snúningi grafsins.

Hann gefur snúningshorn hnitásanna sem eyðir liðnum 𝑥 ⁢ 𝑦 . x ⁢ $xy$y

$AB=0,$ fleygbogi

$AB=-4<0,$ breiðbogi

$AB=6>0,$ sporbaugur

$B^2-4AC=0,$ fleygbogi

$B^2-4AC=0,$ fleygbogi

$B^2-4AC=-96<0,$ sporbaugur

$7{x^{\prime }}^2+9{y^{\prime }}^2-4=0$

$3{x^{\prime }}^2+2x^{\prime }{y}^{\prime }-5{y^{\prime }}^2+1=0$

$\theta ={60}^{\circ },11{x^{\prime }}^2-{y^{\prime }}^2+\sqrt{3}{x}^{\prime }+y^{\prime }-4=0$

$\theta ={-30}^{\circ },21{x^{\prime }}^2+9{y^{\prime }}^2+4x^{\prime }-4\sqrt{3}{y}^{\prime }-6=0$

$\theta \approx {36.9}^{\circ },125{x^{\prime }}^2+6x^{\prime }-42y^{\prime }+10=0$

$\theta ={45}^{\circ },3{x^{\prime }}^2-{y^{\prime }}^2-\sqrt{2}{x}^{\prime }+\sqrt{2}{y}^{\prime }+1=0$

$\frac{\sqrt{2}}{2}\left(x^{\prime }+y^{\prime }\right)=\frac{1}{2}{\left(x^{\prime }-y^{\prime }\right)}^2$

$\frac{{\left(x^{\prime }-y^{\prime }\right)}^2}{8}+\frac{{\left(x^{\prime }+y^{\prime }\right)}^2}{2}=1$

$\frac{{\left(x^{\prime }+y^{\prime }\right)}^2}{2}-\frac{{\left(x^{\prime }-y^{\prime }\right)}^2}{2}=1$

$\frac{\sqrt{3}}{2}{x}^{\prime }-\frac{1}{2}{y}^{\prime }={\left(\frac{1}{2}{x}^{\prime }+\frac{\sqrt{3}}{2}{y}^{\prime }-1\right)}^2$

$\theta ={45}^{\circ }$

$\theta ={60}^{\circ }$

$\theta \approx {36.9}^{\circ }$

$-4\sqrt{6}<k<4\sqrt{6}$

$k=2$

Fleygbogi með $e=1$ og stýrilínu $\frac{3}{4}$ einingum neðan við skautið.

Breiðbogi með $e=2$ og stýrilínu $\frac{5}{2}$ einingum ofan við skautið.

Fleygbogi með $e=1$ og stýrilínu $\frac{3}{10}$ einingum hægra megin við skautið.

Sporbaugur með $e=\frac{2}{7}$ og stýrilínu $2$ einingum hægra megin við skautið.

Breiðbogi með $e=\frac{5}{3}$ og stýrilínu $\frac{11}{5}$ einingum ofan við skautið.

Breiðbogi með $e=\frac{8}{7}$ og stýrilínu $\frac{7}{8}$ einingum hægra megin við skautið.

$25x^2+16y^2-12y-4=0$

$21x^2-4y^2-30x+9=0$

$64y^2=48x+9$

$96y^2-25x^2+110y+25=0$

$3x^2+4y^2-2x-1=0$

$5x^2+9y^2-24x-36=0$

$r=\frac{4}{5+\cos \theta }$

$r=\frac{4}{1+2\sin \theta }$

$r=\frac{1}{1+\cos \theta }$

$r=\frac{7}{8-28\cos \theta }$

$r=\frac{12}{2+3\sin \theta }$

$r=\frac{15}{4-3\cos \theta }$

$r=\frac{3}{3-3\cos \theta }$

$r=\pm \frac{2}{\sqrt{1+\sin \theta \cos \theta }}$

$r=\pm \frac{2}{4\cos \theta +3\sin \theta }$

$\frac{x^2}{5^2}+\frac{y^2}{8^2}=1;$ miðja: $\left(0,0\right);$ hvirfilpunktar: $\left(5,0\right),\left(\mathrm{-5},0\right),\left(0,8\right),\left(0,-8\right);$ brennipunktar: $\left(0,\sqrt{39}\right),\left(0,-\sqrt{39}\right)$

$\frac{{(x+3)}^2}{1^2}+\frac{{(y-2)}^2}{3^2}=1(-3,2);(-2,2),(-4,2),(-3,5),(-3,-1);\left(-3,2+2\sqrt{2}\right),\left(-3,2-2\sqrt{2}\right)$

miðja: $\left(0,0\right);$ hvirfilpunktar: $\left(6,0\right),\left(\mathrm{-6},0\right),\left(0,3\right),\left(0,\mathrm{-3}\right);$ brennipunktar: $\left(3\sqrt{3},0\right),\left(-3\sqrt{3},0\right)$

miðja: $\left(\mathrm{-2},\mathrm{-2}\right);$ hvirfilpunktar: $\left(2,\mathrm{-2}\right),\left(\mathrm{-6},\mathrm{-2}\right),\left(\mathrm{-2},6\right),\left(\mathrm{-2},\mathrm{-10}\right);$ brennipunktar: $\left(\mathrm{-2},\mathrm{-2}+4\sqrt{3},\right),\left(\mathrm{-2},\mathrm{-2}\mathrm{-4}\sqrt{3}\right)$

$\frac{x^2}{25}+\frac{y^2}{16}=1$

$\frac{{\left(y+1\right)}^2}{4^2}-\frac{{\left(x-4\right)}^2}{6^2}=1;$ miðja: $\left(4,\mathrm{-1}\right);$ hvirfilpunktar: $\left(4,3\right),\left(4,\mathrm{-5}\right);$ brennipunktar: $\left(4,\mathrm{-1}+2\sqrt{13}\right),\left(4,\mathrm{-1}-2\sqrt{13}\right)$

$\frac{{\left(x-2\right)}^2}{2^2}-\frac{{\left(y+3\right)}^2}{{\left(2\sqrt{3}\right)}^2}=1;$ miðja: $\left(2,\mathrm{-3}\right);$ hvirfilpunktar: $\left(4,\mathrm{-3}\right),\left(0,\mathrm{-3}\right);$ brennipunktar: $\left(6,\mathrm{-3}\right),\left(\mathrm{-2},\mathrm{-3}\right)$

$\frac{{\left(x-5\right)}^2}{1}-\frac{{\left(y-7\right)}^2}{3}=1$