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# In the figure shown, the circle has center O and radius 50

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In the figure shown, the circle has center O and radius 50 [#permalink]

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03 Jan 2011, 18:18
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In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?

(1) The x-coordinate of point Q is – 30.
(2) The y-coordinate of point Q is – 40.
[Reveal] Spoiler: OA

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Ajit

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Re: figure shown, the circle has center O and radius 50 [#permalink]

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03 Jan 2011, 21:19
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We need to have both the X and Y coordinate to find the distance.
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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04 Jan 2011, 03:03
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In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?

Note that we are told that point Q is on the circle. Also as the radius of the circle is 50 then for any point (x,y) on the circle $$x^2+y^2=50^2$$ (check for more here: math-coordinate-geometry-87652.html) Look at the diagram:
Attachment:

Circle.png [ 13.28 KiB | Viewed 12668 times ]

(1) The x-coordinate of point Q is – 30 --> point Q can be either on the position of Q1 or Q2 on the diagram, but in any case the distance between P and Q is the same --> for point Q: $$(-30)^2+y^2=50^2$$ --> $$y^2=20*80=40^2$$, so $$CQ^2=y^2=40^2$$ (no matter where Q actually is on Q1 or Q2) --> PQ which is the hypotenuse in PQC is equal to $$PQ^2=PC^2+CQ^2=80^2+40^2$$ --> $$PQ=40\sqrt{5}$$. Sufficient.

(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases. Not sufficient.

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Re: figure shown, the circle has center O and radius 50 [#permalink]

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23 Feb 2012, 01:48
How doe we know that P lies on the diameter ? As per question, position of P relative to centre is not known. So how is it assumed that P is on the diameter? Wouldn't it change answer ?
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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23 Feb 2012, 02:24
deepo85 wrote:
How doe we know that P lies on the diameter ? As per question, position of P relative to centre is not known. So how is it assumed that P is on the diameter? Wouldn't it change answer ?

Since the circle is centered at the origin and has the radius 50, then point P(50,0) must lie on a circle.

Hope it's clear.
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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28 Apr 2012, 11:39
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Nice question I got it wrong because i just considered X co-ordinate ,forgot that things could be different for y
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Shikhar

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Re: In the figure shown, the circle has center O and radius 50 [#permalink]

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22 Jul 2013, 22:17
Bumping for review and further discussion.
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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03 Aug 2013, 07:52
Bunuel wrote:
In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?

Note that we are told that point Q is on the circle. Also as the radius of the circle is 50 then for any point (x,y) on the circle $$x^2+y^2=50^2$$ (check for more here: math-coordinate-geometry-87652.html) Look at the diagram:
Attachment:
Circle.png

(1) The x-coordinate of point Q is – 30 --> point Q can be either on the position of Q1 or Q2 on the diagram, but in any case the distance between P and Q is the same --> for point Q: $$(-30)^2+y^2=50^2$$ --> $$y^2=20*80=40^2$$, so $$CQ^2=y^2=40^2$$ (no matter where Q actually is on Q1 or Q2) --> PQ which is the hypotenuse in PQC is equal to $$PQ^2=PC^2+CQ^2=80^2+40^2$$ --> $$PQ=40\sqrt{5}$$. Sufficient.

(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases. Not sufficient.

why cannot the point Q have coordinates (-30,0) in this case point Q will lie on the Diameter of the circle and the distance from point P will be 80, so aren't we getting two cases for A?
What am I missing here? Can anybody assist?
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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03 Aug 2013, 08:01
stne wrote:
Bunuel wrote:
In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?

Note that we are told that point Q is on the circle. Also as the radius of the circle is 50 then for any point (x,y) on the circle $$x^2+y^2=50^2$$ (check for more here: math-coordinate-geometry-87652.html) Look at the diagram:
Attachment:
Circle.png

(1) The x-coordinate of point Q is – 30 --> point Q can be either on the position of Q1 or Q2 on the diagram, but in any case the distance between P and Q is the same --> for point Q: $$(-30)^2+y^2=50^2$$ --> $$y^2=20*80=40^2$$, so $$CQ^2=y^2=40^2$$ (no matter where Q actually is on Q1 or Q2) --> PQ which is the hypotenuse in PQC is equal to $$PQ^2=PC^2+CQ^2=80^2+40^2$$ --> $$PQ=40\sqrt{5}$$. Sufficient.

(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases. Not sufficient.

why cannot the point Q have coordinates (-30,0) in this case point Q will lie on the Diameter of the circle and the distance from point P will be 80, so aren't we getting two cases for A?
What am I missing here? Can anybody assist?

You are forgetting that if Q is (-30,0), then the point will no longer be ON the circle. It will be IN the circle.
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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03 Aug 2013, 09:50
mau5 wrote:
stne wrote:
Bunuel wrote:
In the figure shown, the circle has center O and radius 50, and point P has coordinates (50,0). If point Q (not shown) is on the circle, what is the length of line segment PQ ?

Note that we are told that point Q is on the circle. Also as the radius of the circle is 50 then for any point (x,y) on the circle $$x^2+y^2=50^2$$ (check for more here: math-coordinate-geometry-87652.html) Look at the diagram:
Attachment:
Circle.png

(1) The x-coordinate of point Q is – 30 --> point Q can be either on the position of Q1 or Q2 on the diagram, but in any case the distance between P and Q is the same --> for point Q: $$(-30)^2+y^2=50^2$$ --> $$y^2=20*80=40^2$$, so $$CQ^2=y^2=40^2$$ (no matter where Q actually is on Q1 or Q2) --> PQ which is the hypotenuse in PQC is equal to $$PQ^2=PC^2+CQ^2=80^2+40^2$$ --> $$PQ=40\sqrt{5}$$. Sufficient.

(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases. Not sufficient.

why cannot the point Q have coordinates (-30,0) in this case point Q will lie on the Diameter of the circle and the distance from point P will be 80, so aren't we getting two cases for A?
What am I missing here? Can anybody assist?

You are forgetting that if Q is (-30,0), then the point will no longer be ON the circle. It will be IN the circle.

Correct! Realized this just after I posted it. Although couldn't it have been more clearer if it was mentioned that point Q lies on the circumference of the circle? Just ON the circle can also mean anywhere in the circle,can it not?
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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03 Aug 2013, 09:54
stne wrote:

Correct! Realized this just after I posted it. Although couldn't it have been more clearer if it was mentioned that point Q lies on the circumference of the circle? Just ON the circle can also mean anywhere in the circle,can it not?

No.A point lying on the circumference or a point lying on the circle is the same thing.IF it is on the circle, it can't be anywhere else.
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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03 Aug 2013, 10:01
mau5 wrote:
stne wrote:

Correct! Realized this just after I posted it. Although couldn't it have been more clearer if it was mentioned that point Q lies on the circumference of the circle? Just ON the circle can also mean anywhere in the circle,can it not?

No.A point lying on the circumference or a point lying on the circle is the same thing.IF it is on the circle, it can't be anywhere else.

Let me just cement my understanding,ON and circumference is the same thing, So if it says that a point lies on the circle, it always means that the point lies on the circumference, is that correct.
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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03 Aug 2013, 10:05
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stne wrote:
mau5 wrote:
stne wrote:

Correct! Realized this just after I posted it. Although couldn't it have been more clearer if it was mentioned that point Q lies on the circumference of the circle? Just ON the circle can also mean anywhere in the circle,can it not?

No.A point lying on the circumference or a point lying on the circle is the same thing.IF it is on the circle, it can't be anywhere else.

Let me just cement my understanding,ON and circumference is the same thing, So if it says that a point lies on the circle, it always means that the point lies on the circumference, is that correct.

Yes, that is correct.Any point on the circle is a point on the circumference.
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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03 Aug 2013, 10:19
mau5 wrote:
Yes, that is correct.Any point on the circle is a point on the circumference.

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Re: In the figure shown, the circle has center O and radius 50 [#permalink]

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04 Aug 2013, 00:16
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Hi,

Lets look at this in another way, Equation of Circle: $$x^2 + y^2 = 2500$$

Statement (1): The x-coordinate of point Q is – 30.

Substituting this in $$x^2 + y^2 = 2500$$,
we get y = 40 or -40

Now using distance formula,
dist(P,Q) = $$\sqrt{(x - x')^2 + (y - y')^2}$$
= $$\sqrt{(50 - (-30))^2 + (0 - (40))^2}$$ or = $$\sqrt{(50 - (-30))^2 + (0 - (-40))^2}$$
= $$\sqrt{(6400 + 1600)}$$
= $$\sqrt{8000}$$

So in either case, dist(P,Q) = $$\sqrt{8000}$$. Hence, Sufficient.

Statement (2): The y-coordinate of point Q is – 40.
Substituting this in $$x^2 + y^2 = 2500$$,
we get x = 30 or -30

Now using distance formula,
dist(P,Q) = $$\sqrt{(x - x')^2 + (y - y')^2}$$
= $$\sqrt{(50 - (-30))^2 + (0 - (-40))^2}$$ OR = $$\sqrt{(50 - (30))^2 + (0 - (-40))^2}$$
= $$\sqrt{(6400 + 1600)}$$ OR = $$\sqrt{(400 + 1600)}$$
= $$\sqrt{8000}$$ OR = $$\sqrt{2000}$$

So different answers depending on whether x = -30 or x = 30. Hence, Insufficient.

Correct Ans: A
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Re: In the figure shown, the circle has center O and radius 50 [#permalink]

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04 Aug 2013, 00:24
Hi,

Lets look at this in another way, Equation of Circle: $$x^2 + y^2 = 2500$$

Statement (1): The x-coordinate of point Q is – 30.

Substituting this in $$x^2 + y^2 = 2500$$,
we get y = 40 or -40

Now using distance formula,
dist(P,Q) = $$\sqrt{(x - x')^2 + (y - y')^2}$$
= $$\sqrt{(50 - (-30))^2 + (0 - (40))^2}$$ or = $$\sqrt{(50 - (-30))^2 + (0 - (-40))^2}$$
= $$\sqrt{(6400 + 1600)}$$
= $$\sqrt{8000}$$

So in either case, dist(P,Q) = $$\sqrt{8000}$$. Hence, Sufficient.

Statement (2): The y-coordinate of point Q is – 40.
Substituting this in $$x^2 + y^2 = 2500$$,
we get x = 30 or -30

Now using distance formula,
dist(P,Q) = $$\sqrt{(x - x')^2 + (y - y')^2}$$
= $$\sqrt{(50 - (-30))^2 + (0 - (-40))^2}$$ OR = $$\sqrt{(50 - (30))^2 + (0 - (-40))^2}$$
= $$\sqrt{(6400 + 1600)}$$ OR = $$\sqrt{(400 + 1600)}$$
= $$\sqrt{8000}$$ OR = $$\sqrt{2000}$$

So different answers depending on whether x = -30 or x = 30. Hence, Insufficient.

Correct Ans: A

+1 ,Simple explanation.
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Re: figure shown, the circle has center O and radius 50 [#permalink]

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15 Apr 2014, 00:46
HI Bunnel,

I am not clear about second statement.

(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases. Not sufficient.

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Posts: 39593
Re: figure shown, the circle has center O and radius 50 [#permalink]

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15 Apr 2014, 01:12
HI Bunnel,

I am not clear about second statement.

(2) The y-coordinate of point Q is – 40 --> point Q can be either on the position of Q2 or Q3 on the diagram, and the distance between P and Q will be different for theses cases. Not sufficient.

Look at the diagram below:
Attachment:

Untitled.png [ 11.5 KiB | Viewed 7157 times ]
We are told that the y-coordinate of point Q is -40, so it's either on the position of Q2 or Q3. As you can see these different positions give different distances for PQ (blue and green lines).

Hope it's clear.
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Re: In the figure shown, the circle has center O and radius 50 [#permalink]

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21 Jul 2014, 13:47
Hi,

Lets look at this in another way, Equation of Circle: $$x^2 + y^2 = 2500$$

Statement (1): The x-coordinate of point Q is – 30.

Substituting this in $$x^2 + y^2 = 2500$$,
we get y = 40 or -40

Now using distance formula,
dist(P,Q) = $$\sqrt{(x - x')^2 + (y - y')^2}$$
= $$\sqrt{(50 - (-30))^2 + (0 - (40))^2}$$ or = $$\sqrt{(50 - (-30))^2 + (0 - (-40))^2}$$
= $$\sqrt{(6400 + 1600)}$$
= $$\sqrt{8000}$$

So in either case, dist(P,Q) = $$\sqrt{8000}$$. Hence, Sufficient.

Statement (2): The y-coordinate of point Q is – 40.
Substituting this in $$x^2 + y^2 = 2500$$,
we get x = 30 or -30

Now using distance formula,
dist(P,Q) = $$\sqrt{(x - x')^2 + (y - y')^2}$$
= $$\sqrt{(50 - (-30))^2 + (0 - (-40))^2}$$ OR = $$\sqrt{(50 - (30))^2 + (0 - (-40))^2}$$
= $$\sqrt{(6400 + 1600)}$$ OR = $$\sqrt{(400 + 1600)}$$
= $$\sqrt{8000}$$ OR = $$\sqrt{2000}$$

So different answers depending on whether x = -30 or x = 30. Hence, Insufficient.

Correct Ans: A

Do we plug the value for x and y into x and y? --> meaning for stmt 2 do we do x^2 + -40^2 = 2500?
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Re: In the figure shown, the circle has center O and radius 50 [#permalink]

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04 Feb 2016, 19:06
oh man..just made a crucial mistake and mistook x for y and y for x.

if we know X, we then can have 2 points - positive Y and negative Y. in either case, the distance from Q to P will be the same, since the points would be a reflection, and the distance would be the same.

knowing for X - we can answer the question.
knowing for Y - we can't answer, since X can be either positive or negative. 2 different values.
thus, A is sufficient.
Re: In the figure shown, the circle has center O and radius 50   [#permalink] 04 Feb 2016, 19:06

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