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The equation of a circle in the x-y coordinate plane is x^2+y^2=25.

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The equation of a circle in the x-y coordinate plane is x^2+y^2=25.  [#permalink]

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New post 29 Nov 2018, 02:47
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[Math Revolution GMAT math practice question]

The equation of a circle in the \(x-y\) coordinate plane is \(x^2+y^2=25\). How many points of the form (\(a,b\)), where \(a\) and \(b\) are integers, lie on this circle?

\(A. 4\)
\(B. 6\)
\(C. 8\)
\(D. 10\)
\(E. 12\)

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Re: The equation of a circle in the x-y coordinate plane is x^2+y^2=25.  [#permalink]

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New post 29 Nov 2018, 04:11
MathRevolution wrote:
[Math Revolution GMAT math practice question]

The equation of a circle in the \(x-y\) coordinate plane is \(x^2+y^2=25\). How many points of the form (\(a,b\)), where \(a\) and \(b\) are integers, lie on this circle?

\(A. 4\)
\(B. 6\)
\(C. 8\)
\(D. 10\)
\(E. 12\)



Hi..

The equation \(x^2+y^2=25......x^2+y^2=5^2\) is an equation of a circle with its center at the origin of the coordinates and the radius as 5..

What are the coordinates (a,b) and radius 5..
they are the sides of a right angled triangle with hypotenuse 5.. 3-4-5 is a triplet which we will be looking at
so we can have coordinates (|3|,|4|)... here 3 and 4 both can take positive and negative values, so 2*2=4------- (3,4);(-3,4);(-3,-4);(3,-4)
similarly four ways for (|4|,|3|)
The points (0,5), (0,-5), (5,0) and (-5,0) are other 4 ways.

Thus 4+4+4=12 ways

E
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Re: The equation of a circle in the x-y coordinate plane is x^2+y^2=25.  [#permalink]

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New post 29 Nov 2018, 07:57
MathRevolution wrote:
[Math Revolution GMAT math practice question]

The equation of a circle in the \(x-y\) coordinate plane is \(x^2+y^2=25\). How many points of the form (\(a,b\)), where \(a\) and \(b\) are integers, lie on this circle?

\(A. 4\)
\(B. 6\)
\(C. 8\)
\(D. 10\)
\(E. 12\)

\(?\,\,\,:\,\,\,\,\# \,\,\left( {x,y} \right)\,\,\,{\rm{integer}}\,\,{\rm{coordinates}}\,\,{\rm{solutions}}\,\,{\rm{for}}\,\,\,{x^2} + {y^2} = 25\)


\(\left| x \right| = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {x,y} \right) = \left( {0,5} \right)\,\,\,{\rm{or}}\,\,\,\left( {x,y} \right) = \left( {0, - 5} \right)\)

\(\left| x \right| = 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{y^2} = 24 \ne \,\,{\rm{perfect}}\,\,{\rm{square}}\,\,\,\, \Rightarrow \,\,\,\,{\rm{no}}\,\,{\rm{solutions}}\,\)

\(\left| x \right| = 2\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{y^2} = 21 \ne \,\,{\rm{perfect}}\,\,{\rm{square}}\,\,\,\, \Rightarrow \,\,\,\,{\rm{no}}\,\,{\rm{solutions}}\,\,\,\)

\(\left| x \right| = 3\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{y^2} = 16\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,\left( {x,y} \right) = \left( {3,4} \right)\,\,\,{\rm{or}}\,\,\,\left( {x,y} \right) = \left( {3, - 4} \right) \hfill \cr
\,\,\,{\rm{or}} \hfill \cr
\,\,\left( {x,y} \right) = \left( { - 3,4} \right)\,\,\,{\rm{or}}\,\,\,\left( {x,y} \right) = \left( { - 3, - 4} \right) \hfill \cr} \right.\)

\(\left| x \right| = 4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{y^2} = 9\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,\left( {x,y} \right) = \left( {4,3} \right)\,\,\,{\rm{or}}\,\,\,\left( {x,y} \right) = \left( {4, - 3} \right) \hfill \cr
\,\,\,{\rm{or}} \hfill \cr
\,\,\left( {x,y} \right) = \left( { - 4,3} \right)\,\,\,{\rm{or}}\,\,\,\left( {x,y} \right) = \left( { - 4, - 3} \right) \hfill \cr} \right.\,\,\)

\(\left| x \right| = 5\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{y^2} = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {x,y} \right) = \left( {5,0} \right)\,\,\,{\rm{or}}\,\,\,\left( {x,y} \right) = \left( { - 5,0} \right)\)


\(? = 12\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: The equation of a circle in the x-y coordinate plane is x^2+y^2=25.  [#permalink]

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New post 02 Dec 2018, 19:01
=>

We need to find all pairs of integers (\(a,b\)) such that \(a^2 + b^2 = 25\). These are:
\((5,0), (4,3), (3,4), (0,5), (-3,4), (-4,3), (-5,0),(-4,-3), (-3,-4), (0,-5), (3,-4)\) and \((4,-3)\).
Thus, there are \(12\) such points that lie on the circle.

Therefore, the answer is E.
Answer: E
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Re: The equation of a circle in the x-y coordinate plane is x^2+y^2=25.   [#permalink] 02 Dec 2018, 19:01
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