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Solution 4.1:6c

From Förberedande kurs i matematik 1

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m (Lösning 4.1:6c moved to Solution 4.1:6c: Robot: moved page)
Current revision (11:45, 27 September 2008) (edit) (undo)
 
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What we need to do is to rewrite the equation in the standard form
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<center> [[Image:4_1_6c-1(2).gif]] </center>
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<math>\left( x-a \right)^{2}+\left( y-b \right)^{2}=r^{2}</math>
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because then we can read off the circle's centre
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<math>\left( a \right.,\left. b \right)</math>
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and radius,
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<math>r</math>.
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In our case, we need only take out the factor
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<math>~\text{3}</math>
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from the brackets on the left-hand side
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<math>\begin{align}
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& \left( 3x-1 \right)^{2}+\left( 3y+7 \right)^{2}=3^{2}\left( x-\frac{1}{3} \right)^{2}+3^{2}\left( y+\frac{7}{3} \right)^{2} \\
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& =9\left( x-\frac{1}{3} \right)^{2}+9\left( y+\frac{7}{3} \right)^{2} \\
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\end{align}</math>
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and then divide both sides by
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<math>\text{9}</math>
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, so as to get the equation in the desired form:
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<math>\left( x-\frac{1}{3} \right)^{2}+\left( y+\frac{7}{3} \right)^{2}=\frac{10}{9}</math>
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Because the right-hand side can be written as
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<math>\left( \sqrt{\frac{10}{9}} \right)^{2}</math>
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and the term
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<math>\left( y+\frac{7}{3} \right)^{2}</math>
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as
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<math></math>
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<math>\left( y-\left( -\frac{7}{3} \right) \right)^{2}</math>, the equation describes a circle with its centre at
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<math>\left( \frac{1}{3} \right.,\left. -\frac{7}{3} \right)</math>
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and radius
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<math>\sqrt{\frac{10}{9}}=\frac{\sqrt{10}}{3}</math>
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<center> [[Image:4_1_6c-2(2).gif]] </center>
<center> [[Image:4_1_6c-2(2).gif]] </center>
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Current revision

What we need to do is to rewrite the equation in the standard form


x−a2+y−b2=r2 


because then we can read off the circle's centre ab  and radius, r.

In our case, we need only take out the factor  3 from the brackets on the left-hand side


3x−12+3y+72=32x−312+32y+372=9x−312+9y+372


and then divide both sides by 9 , so as to get the equation in the desired form:


x−312+y+372=910 

Because the right-hand side can be written as 9102  and the term y+372  as

y−−372 , the equation describes a circle with its centre at 31−37  and radius 910=310 

Image:4_1_6c-2(2).gif
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