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

Question

Circle.png 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.

Bunuel · Accepted Answer

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 27622 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.

Answer: A.

prab · Answer

We need to have both the X and Y coordinate to find the distance.

deepo85 · Answer

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 ?

Bunuel · 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.

shikhar · Answer

Nice question I got it wrong because i just considered X co-ordinate ,forgot that things could be different for y

stne · Answer

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?

mau5 · Answer

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.

stne · Answer

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?

mau5 · Answer

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.

stne · Answer

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.

mau5 · Answer

Yes, that is correct.Any point on the circle is a point on the circumference.

stne · Answer

Thank you +1, it was an eyeopener, had been unsure about this.

kumar23badgujar · Answer

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

kewldude762 · Answer

+1 ,Simple explanation. :-D

PathFinder007 · Answer

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.

Bunuel · Answer

Look at the diagram below: Untitled.png 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.

sagnik2422 · Answer

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?

mvictor · Answer

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.

longhaul123 · Answer

Can someone reason the statement 2 as to how is it different from statement 1 and what if given in statement 2 would be sufficient?

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

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