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In the figure shown, the circle has center O and radius 50 [#permalink]
03 Jan 2011, 17:18

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Difficulty:

95% (hard)

Question Stats:

39% (02:19) correct
61% (01:14) wrong based on 202 sessions

Attachment:

Circle.png [ 14.99 KiB | Viewed 8575 times ]

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.

Re: figure shown, the circle has center O and radius 50 [#permalink]
04 Jan 2011, 02:03

6

This post received KUDOS

Expert's post

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 8548 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.

Re: figure shown, the circle has center O and radius 50 [#permalink]
23 Feb 2012, 00:48

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 ?

Re: figure shown, the circle has center O and radius 50 [#permalink]
23 Feb 2012, 01:24

Expert's post

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.

Re: figure shown, the circle has center O and radius 50 [#permalink]
03 Aug 2013, 06:52

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.

Answer: A.

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? _________________

Re: figure shown, the circle has center O and radius 50 [#permalink]
03 Aug 2013, 07:01

Expert's post

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.

Answer: A.

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. _________________

Re: figure shown, the circle has center O and radius 50 [#permalink]
03 Aug 2013, 08:50

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.

Answer: A.

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? _________________

Re: figure shown, the circle has center O and radius 50 [#permalink]
03 Aug 2013, 08:54

Expert's post

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. _________________

Re: figure shown, the circle has center O and radius 50 [#permalink]
03 Aug 2013, 09:01

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. _________________

Re: figure shown, the circle has center O and radius 50 [#permalink]
03 Aug 2013, 09:05

1

This post received KUDOS

Expert's post

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. _________________

Re: figure shown, the circle has center O and radius 50 [#permalink]
14 Apr 2014, 23:46

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.

Re: figure shown, the circle has center O and radius 50 [#permalink]
15 Apr 2014, 00:12

Expert's post

pawankumargadiya wrote:

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.

Please clarify.

Look at the diagram below:

Attachment:

Untitled.png [ 11.5 KiB | Viewed 3060 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).

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