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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   85% (hard)

Question Stats: 36% (01:28) correct 64% (02:07) wrong based on 75 sessions

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Does the curve $$(x - a)^2 + (y - b)^2 = 16$$ intersect the $$Y$$ axis?

(1) $$a^2 + b^2 \gt 16$$

(2) $$a = |b| + 5$$

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Math Expert V
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Official Solution:

THEORY

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: $$x^2+y^2=r^2$$

For more check: http://gmatclub.com/forum/math-coordina ... 87652.html

BACK TO THE ORIGINAL QUESTION

Does the curve $$(x - a)^2 + (y - b)^2 = 16$$ intersect the $$y$$ axis?

Curve of $$(x - a)^2 + (y - b)^2 = 16$$ is a circle centered at the point $$\text{(a, b)}$$ and has a radius of $$\sqrt{16}=4$$. Now, if $$a$$, the x-coordinate of the center, is more than 4 or less than -4 then the radius of the circle, which is 4, won't be enough for curve to intersect with Y axis. So basically the question asks whether $$|a| \gt 4$$: if it is, then the answer will be NO: the curve does not intersect with Y axis and if it's not, then the answer will be YES: the curve intersects with Y axis.

(1) $$a^2 + b^2 \gt 16$$. Clearly insufficient as $$|a|$$ may or may not be more than 4.

(2) $$a = |b| + 5$$. As the least value of absolute value (in our case $$|b|$$) is zero then the least value of $$a$$ will be 5, so in any case $$|a| \gt 4$$, which means that the circle does not intersect the Y axis. Sufficient.

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Hi Bunuel I do not understand this "Now, if a, the x-coordinate of the center, is more than 4 or less than -4 then the radius of the circle, which is 4, won't be enough for curve to intersect with Y axis" Can you explain it for me ? many thanks :D !
Intern  Joined: 01 Oct 2014
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this is great.
Math Expert V
Joined: 02 Sep 2009
Posts: 58464

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langtuprovn2007 wrote:
Hi Bunuel I do not understand this "Now, if a, the x-coordinate of the center, is more than 4 or less than -4 then the radius of the circle, which is 4, won't be enough for curve to intersect with Y axis" Can you explain it for me ? many thanks :D !

This should be easy if you draw it. If the center is more than 4 units far from y-axis then the radius of 4 won't be enough to reach y-axis:
Attachment:
Untitled.png

>> !!!

You do not have the required permissions to view the files attached to this post.

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Manager  Joined: 06 Mar 2014
Posts: 220
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GMAT Date: 04-30-2015

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Bunuel wrote:
langtuprovn2007 wrote:
Hi Bunuel I do not understand this "Now, if a, the x-coordinate of the center, is more than 4 or less than -4 then the radius of the circle, which is 4, won't be enough for curve to intersect with Y axis" Can you explain it for me ? many thanks :D !

This should be easy if you draw it. If the center is more than 4 units far from y-axis then the radius of 4 won't be enough to reach y-axis:
Attachment:
Untitled.png

This is great, however i am unable to visualize the case when a co-ordinate is less than -4. Then how exactly will the circle not be able to touch the y axis?
Math Expert V
Joined: 02 Sep 2009
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earnit wrote:
Bunuel wrote:
langtuprovn2007 wrote:
Hi Bunuel I do not understand this "Now, if a, the x-coordinate of the center, is more than 4 or less than -4 then the radius of the circle, which is 4, won't be enough for curve to intersect with Y axis" Can you explain it for me ? many thanks :D !

This should be easy if you draw it. If the center is more than 4 units far from y-axis then the radius of 4 won't be enough to reach y-axis:
Attachment:
Untitled.png

This is great, however i am unable to visualize the case when a co-ordinate is less than -4. Then how exactly will the circle not be able to touch the y axis?

Check below:
Attachment:
Untitled.png

>> !!!

You do not have the required permissions to view the files attached to this post.

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Retired Moderator P
Status: The best is yet to come.....
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I think this question is good and helpful.
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Hasan Mahmud
Intern  Joined: 22 Aug 2014
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The main theme is whether the circle intersects Y axis or not. So, it depends on 'a'.
First, try to answer without (1) or (2). Can you say the circle intersects Y axis? No. It depends on a. If a is closer enough to Y axis so that 4 unit distance is enough to touch. So, 'a' must be at least within 4 distance or /a/=4 or /a/<4
(1) not sufficient because a2>16-b2. Depending on b, a can be either greater than 4(not touch Y) or less than 4(touch Y). Since it fails to justify a definite answer to the original question, it is not sufficient.
(2) Forget about 1. (2) tells that in any situation or even at the least value of b, 'a' must be at least 5. So, radius 4 is not enough to touch Y axis from 5 unit distance(minimum scenario). So sufficient, and the answer is 'No' to the question. B
Intern  Joined: 08 Jul 2015
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I used algebra approach for this question.

So the circle will intersect with y-axis at point $$(0,y_0)$$.

Substitute into the question we have: $$a^2 + (y_0 - b)^2 = 16 <-> a^2 - 16 + (y_0 - b)^2 = 0$$
--> so the question is whether this equation have real root $$y_0$$ or not.

(1) Lead to: $$a^2 - 16$$ can be >0 or <0 --> insufficient
(2) Lead to: $$a^2 - 16 >0 -> a^2 - 16 + (y_0 - b)^2 > 0$$: equation has no real root $$y_0$$ --> no intersect: sufficient.
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I think this is a high-quality question and I agree with explanation. This question really shows why visualization is important in geometry. Awesome Question. Really very thoughtful.
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I think this is a high-quality question and I agree with explanation. Very easy question if you can visualize the picture. That's why I love math  Re M09-03   [#permalink] 05 Mar 2019, 08:07
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