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

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.

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

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?

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.

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

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.

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: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.

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?

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.

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?