Max@Math Revolution 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.
