https://peda.net/id/fe736f4ace2 2019-09-03T12:41:24+03:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/8208667ece3 2019-09-03T14:10:59+03:00 muutetaan ympyröiden yhtälöt keskipistemuotoisiksi, jotta nähdään niiden keskipiste ja säde<br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2-6x-6y%3D0" alt="x^2+y^2-6x-6y=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-3%5Cright)%5E2%2B%5Cleft(y-3%5Cright)%5E2%3D18" alt="\left(x-3\right)^2+\left(y-3\right)^2=18"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(3%7B%2C%7D%5C%203%5Cright)%7B%2C%7D%5C%20r%3D3%5Csqrt%7B2%7D" alt="\left(3{,}\ 3\right){,}\ r=3\sqrt{2}"/><br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2-4x-6y-19%3D0" alt="x^2+y^2-4x-6y-19=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-2%5Cright)%5E2%2B%5Cleft(y-3%5Cright)%5E2%3D23" alt="\left(x-2\right)^2+\left(y-3\right)^2=23"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(2%7B%2C%7D%5C%203%5Cright)%7B%2C%7D%5C%20r%3D%5Csqrt%7B23%7D" alt="\left(2{,}\ 3\right){,}\ r=\sqrt{23}"/><br/> <br/> ympyrät sijaitsevat sisäkkäin, niiden keskipisteiden etäisyyksien erotus on pienempi kuin säteiden erotus 2019-09-03T14:10:59+03:00 https://peda.net/id/45f63220ce3 2019-09-03T14:02:08+03:00 muutetaan ympyröiden yhtälöt keskipistemuotoisiksi, jotta voidaan laskea keskipisteiden välisen etäisyyden ja säteiden erotus<br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2%2B2x-6y%2B5%3D0" alt="x^2+y^2+2x-6y+5=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%2B1%5Cright)%5E2%2B%5Cleft(y-3%5Cright)%5E2%3D5" alt="\left(x+1\right)^2+\left(y-3\right)^2=5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(-1%7B%2C%7D%5C%203%5Cright)%7B%2C%7D%5C%20r%3D%5Csqrt%7B5%7D" alt="\left(-1{,}\ 3\right){,}\ r=\sqrt{5}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2-14x-4y%2B33%3D0" alt="x^2+y^2-14x-4y+33=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-7%5Cright)%5E2%2B%5Cleft(y-2%5Cright)%5E2%3D20" alt="\left(x-7\right)^2+\left(y-2\right)^2=20"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(7%7B%2C%7D%5C%202%5Cright)%7B%2C%7D%5C%20r%3D2%5Csqrt%7B5%7D" alt="\left(7{,}\ 2\right){,}\ r=2\sqrt{5}"/></div> <div> </div> <div>pisteiden välinen etäisyys</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B%5Cleft(x_1-x_2%5Cright)%5E2%2B%5Cleft(y_1-y_2%5Cright)%5E2%7D" alt="\sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B64%2B1%7D%3D%5Csqrt%7B65%7D" alt="\sqrt{64+1}=\sqrt{65}"/></div> <div>ympyröiden välinen etäisyys on pisteiden välisen etäisyyden ja ympyröiden säteiden erotus</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%7B65%7D-3%5Csqrt%7B5%7D" alt="\sqrt{65}-3\sqrt{5}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Capprox1%7B%2C%7D3540...%5Cright)" alt="\left(\approx1{,}3540...\right)"/></div> 2019-09-03T14:02:08+03:00 https://peda.net/id/7f63c506ce3 2019-09-03T13:49:30+03:00 muutetaan ympyröiden yhtälöt keskipistemuotoisiksi, jotta saadaan niiden keskipisteet selville<br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2-2x-4y%2B1%3D0" alt="x^2+y^2-2x-4y+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-1%5Cright)%5E2%2B%5Cleft(y-2%5Cright)%5E2%3D4" alt="\left(x-1\right)^2+\left(y-2\right)^2=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(1%7B%2C%7D%5C%202%5Cright)" alt="\left(1{,}\ 2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2-14x%2B2y%2B49%3D0" alt="x^2+y^2-14x+2y+49=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-7%5Cright)%5E2%2B%5Cleft(y%2B1%5Cright)%5E2%3D1" alt="\left(x-7\right)^2+\left(y+1\right)^2=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(7%7B%2C%7D%5C%20-1%5Cright)" alt="\left(7{,}\ -1\right)"/><br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y-y_0%3Dk%5Cleft(x-x_0%5Cright)" alt="y-y_0=k\left(x-x_0\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D%3D%5Cfrac%7B2-%5Cleft(-1%5Cright)%7D%7B1-7%7D%3D-%5Cfrac%7B3%7D%7B6%7D%3D-%5Cfrac%7B1%7D%7B2%7D" alt="k=\frac{\Delta y}{\Delta x}=\frac{2-\left(-1\right)}{1-7}=-\frac{3}{6}=-\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y-2%3D-%5Cfrac%7B1%7D%7B2%7D%5Cleft(x-1%5Cright)" alt="y-2=-\frac{1}{2}\left(x-1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D-%5Cfrac%7B1%7D%7B2%7Dx%2B2%5C%20%5Cfrac%7B1%7D%7B2%7D" alt="y=-\frac{1}{2}x+2\ \frac{1}{2}"/><br/> <br/> <a href="https://peda.net/p/oskari.lahtinen/mag/4yyym/428/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/mag/4yyym/428/sieppaa-png:file/photo/375ba6967cc0192e3a7f7ff3bf45b57fc2d745c2/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a> 2019-09-03T13:49:26+03:00 https://peda.net/id/4254c598ce3 2019-09-03T13:11:56+03:00 <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B8x%2B16%3D%5Cleft(x%2B4%5Cright)%5E2" alt="x^2+8x+16=\left(x+4\right)^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%5E2-10y%2B25%3D%5Cleft(x-5%5Cright)%5E2" alt="y^2-10y+25=\left(x-5\right)^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2%2B8x-10y-8%3D0" alt="x^2+y^2+8x-10y-8=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B2x%5Ccdot4%2B...%2By%5E2-2y%5Ccdot5%2B...%3D8" alt="x^2+2x\cdot4+...+y^2-2y\cdot5+...=8"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B2x%5Ccdot4%2B16%2By%5E2-2y%5Ccdot5%2B25%3D8%2B16%2B25" alt="x^2+2x\cdot4+16+y^2-2y\cdot5+25=8+16+25"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%2B4%5Cright)%5E2%2B%5Cleft(y-5%5Cright)%5E2%3D49" alt="\left(x+4\right)^2+\left(y-5\right)^2=49"/> 2019-09-03T13:11:56+03:00 https://peda.net/id/8273c170ce3 2019-09-03T13:06:34+03:00 <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%2B3%5Cright)%5E2" alt="\left(x+3\right)^2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2B6x%2B9" alt="x^2+6x+9"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x-2%5Cright)%5E2%2B%5Cleft(y-1%5Cright)%5E2%3D5" alt="\left(x-2\right)^2+\left(y-1\right)^2=5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%2By%5E2-4x-2x%3D0" alt="x^2+y^2-4x-2x=0"/></div> 2019-09-03T13:06:34+03:00