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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} \\
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\end{align}</math>
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 +
 +
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>
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{{NAVCONTENT_START}}
{{NAVCONTENT_START}}
[[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 2−1  must equal the circle's radius, r. Thus, we can obtain the circle's radius by calculating the distance between −11  and 2−1  using the distance formula:


r=2−−12+−1−12=32+−22=9+4=13 


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


x−22+y−−12=132 


which the same as


x−22+y+12=13 

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


x−a2+y−b2=r2 

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