How many points of intersection does the curve x^2 + y^2 = 4 : GMAT Problem Solving (PS)
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# How many points of intersection does the curve x^2 + y^2 = 4

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How many points of intersection does the curve x^2 + y^2 = 4 [#permalink]

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13 Sep 2010, 13:26
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How many points of intersection does the curve x^2 + y^2 = 4 have with line x+y =2 ?

A. 0
B. 1
C. 2
D. 3
E. 4

did not understand the explanation.

Curve $$x^2 + y^2 = 4$$ is a circle with radius 2 and the center at the origin. A line cannot have more than 2 points of intersection with a circle, so we can eliminate choices D and E. To answer the question, we can either draw a sketch or solve the system of equations. The second approach gives$$(2-y)^2 + y^2 = 4$$or $$2y^2 - 4y + 4 = 4$$ or $$y^2 - 2y = 0$$ from where y=0 and y=2. Thus, the line and the circle intersect at points (2,0) and (0,2).
[Reveal] Spoiler: OA
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Re: did not understand the explanation [#permalink]

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13 Sep 2010, 13:45
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seekmba wrote:
did not understand the explanation.

How many points of intersection does the curve $$x^2 + y^2 = 4$$ have with line x+y =2 ?

0
1
2
3
4

Curve $$x^2 + y^2 = 4$$ is a circle with radius 2 and the center at the origin. A line cannot have more than 2 points of intersection with a circle, so we can eliminate choices D and E. To answer the question, we can either draw a sketch or solve the system of equations. The second approach gives$$(2-y)^2 + y^2 = 4$$or $$2y^2 - 4y + 4 = 4$$ or $$y^2 - 2y = 0$$ from where y=0 and y=2. Thus, the line and the circle intersect at points (2,0) and (0,2).

Can you please specify what part of the solution didn't you understand?

Anyway:
In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that:
$$(x-a)^2+(y-b)^2=r^2$$

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram above, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x-a and y-b.

If the circle is centered at the origin (0, 0), then the equation simplifies to:
$$x^2+y^2=r^2$$

So $$x^2 + y^2 = 4$$ is the equation of circle centered at the origin and with radius equal to 2. Now, $$y=2-x$$ is the equation of a line with x-intercept (2, 0) and y-intercept (0,2) these points are also the intercepts of given circle with X and Y axis hence at these points line and circle intersect each other.
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Re: did not understand the explanation [#permalink]

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13 Sep 2010, 14:00
Thanks so much Bunuel. I was not able to picture the solution given. It makes total sense now.
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Re: did not understand the explanation [#permalink]

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13 Sep 2010, 16:16
In circle and line, I beleieve answer could be just 1(tangent) or 2(line). I am not able to recollect what was the method to know if line is tanget to circle or not. Anyone?
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Re: did not understand the explanation [#permalink]

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13 Sep 2010, 16:50
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saxenashobhit wrote:
In circle and line, I beleieve answer could be just 1(tangent) or 2(line). I am not able to recollect what was the method to know if line is tanget to circle or not. Anyone?

Actually there is a third case: when circle and line don't have any points of intersection. As for the solutions: well if it's an easy case, for example if line is $$y=2$$ we can say that it's tangent to the circle right away or if line is $$y=5$$ we can say right away that they don' intersect at all.

For harder cases you can use the approach used in initial post: $$y=x-2$$, substitute $$x$$ by $$y$$ (or vise-versa) in $$x^2+y^2=4$$ and then solve for $$y$$ (this value will be $$y$$ coordinate of the intersection point(s)). If you'll get one solution for $$y$$ it would mean that line is tangent to circle (as you'll get one point (x,y)), if you'll get two solutions for $$y$$ it would mean that line has two intersection points with circle (as you'll get two points (x,y)) and if you'll get no solution for $$y$$ it would mean that line has no intersection point with circle.

Hope it's clear.
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Re: did not understand the explanation [#permalink]

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27 Sep 2010, 06:19
I think this should be included in the circles chapter of the math book
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Re: How many points of intersection does the curve x^2 + y^2 = 4 [#permalink]

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06 Aug 2014, 04:07
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Re: How many points of intersection does the curve x^2 + y^2 = 4 [#permalink]

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21 Feb 2016, 15:08
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Re: How many points of intersection does the curve x^2 + y^2 = 4   [#permalink] 21 Feb 2016, 15:08
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