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Solution 4.1:5b

From Förberedande kurs i matematik 1

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m (Lösning 4.1:5b moved to Solution 4.1:5b: Robot: moved page)
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 +
If the circle is to contain the point
 +
<math>\left( -1 \right.,\left. 1 \right)</math>, then that point's distance away from the centre
 +
<math>\left( 2 \right.,\left. -1 \right)</math>
 +
must equal the circle's radius,
 +
<math>r</math>. Thus, we can obtain the circle's radius by calculating the distance between
 +
<math>\left( -1 \right.,\left. 1 \right)</math>
 +
and
 +
<math>\left( 2 \right.,\left. -1 \right)</math>
 +
using the distance formula:
 +
 +
 +
<math>\begin{align}
 +
& r=\sqrt{\left( 2-\left( -1 \right) \right)^{2}+\left( -1-1 \right)^{2}}=\sqrt{3^{2}+\left( -2 \right)^{2}} \\
 +
& =\sqrt{9+4}=\sqrt{13} \\
 +
\end{align}</math>
 +
 +
 +
When we know the circle's centre and its radius, we can write the equation of the circle,
 +
 +
 +
<math>\left( x-2 \right)^{2}+\left( y-\left( -1 \right) \right)^{2}=\left( \sqrt{13} \right)^{2}</math>
 +
 +
 +
which the same as
 +
 +
 +
<math>\left( x-2 \right)^{2}+\left( y+1 \right)^{2}=13</math>
 +
{{NAVCONTENT_START}}
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[[Image:4_1_5_b-1(2).gif|center]]
[[Image:4_1_5_b-1(2).gif|center]]
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<center> [[Image:4_1_5b-1(2).gif]] </center>
 
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{{NAVCONTENT_STOP}}
 
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{{NAVCONTENT_START}}
 
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<center> [[Image:4_1_5b-2(2).gif]] </center>
 
{{NAVCONTENT_STOP}}
{{NAVCONTENT_STOP}}
 +
 +
NOTE: A circle having its centre at
 +
<math>\left( a \right.,\left. b \right)</math>
 +
and radius
 +
<math>r</math>
 +
has the equation
 +
 +
 +
<math>\left( x-a \right)^{2}+\left( y-b \right)2=r^{2}</math>

Revision as of 11:20, 27 September 2008

If the circle is to contain the point 11 , then that point's distance away from the centre 21  must equal the circle's radius, r. Thus, we can obtain the circle's radius by calculating the distance between 11  and 21  using the distance formula:


r=212+112=32+22=9+4=13 


When we know the circle's centre and its radius, we can write the equation of the circle,


x22+y12=132 


which the same as


x22+y+12=13 

NOTE: A circle having its centre at ab  and radius r has the equation


xa2+yb2=r2