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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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29 Nov 2018, 01:47
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65% (hard)

Question Stats:

44% (01:12) correct 56% (01:06) wrong based on 39 sessions

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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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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Expert Joined: 02 Aug 2009 Posts: 7108 Re: The equation of a circle in the x-y coordinate plane is x^2+y^2=25. [#permalink] ### Show Tags 29 Nov 2018, 03:11 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 _________________ 1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html GMAT online Tutor GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 546 Re: The equation of a circle in the x-y coordinate plane is x^2+y^2=25. [#permalink] ### Show Tags 29 Nov 2018, 06:57 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. _________________ Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount! Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: The equation of a circle in the x-y coordinate plane is x^2+y^2=25. [#permalink] ### Show Tags 02 Dec 2018, 18:01 => 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: The equation of a circle in the x-y coordinate plane is x^2+y^2=25. &nbs [#permalink] 02 Dec 2018, 18:01
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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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