P is a point (not shown in the figure) in the plane containing O, R, a

Question

https://gmatclub.com/forum/download/file.php?id=54720 P is a point (not shown in the figure) in the plane containing O, R, and Q where O is the center of the circle and R and Q are points on the circle. If r is the radius of the circle, is P inside the circle? (1) PR > PO (2) PR + PQ = 5r DS21262 1.png

nick1816 · Answer

Statement 1- Untitled.png Insufficient Statement 2- Untitled.png If P lies inside the circle, Maximum value of PR and PQ is always less than 4r. Sufficient

Kali123 · Answer

VeritasKarishma .. Could you provide a better solution?

TestPrepUnlimited · Answer

Statement 1: 
P must be further away from R than O, we can have the two cases showed in the attachment. Insufficient.

Statement 2: 
While keeping P within the circle, PR is at most 2r. PQ is also at most 2r. Considering that, even if we add the maximums directly we cannot hit 5r. While keeping P inside the circle, the maximum of PR + PQ involves an optimization problem and it is certainly less than 4r. Therefore P must be outside the circle. Sufficient.

Ans: B

KarishmaB · Answer

(1) PR > PO

Consider point P in the top half of the circle somewhere. PR > PO.
Now move the point P vertically up outside the circle. Still PR > PO.
So P may be inside or outside the circle - not sufficient

(2) PR + PQ = 5r

If two points are inside or at the circle, maximum distance between them will be 2r (the diameter). For any points Q and R on the circle, the maximum sum of distance of P from Q and R will be 2 * 2r = 4r.
If the sum of the distances is 5r, P MUST be outside the circle. 
Sufficient

Answer (B)

Basshead · Answer

(1) One way to visualize this statement is that Point P is closer to Point O than Point R. This could certainly be possible inside the circle and outside the circle. INSUFFICIENT.

(2)  PR + PQ = 5r

Given the radius = r
diameter = 2r

Since we're told PR + PQ = 5r, we can conclude with certainty that P is NOT in the circle because the diameter is 2r. SUFFICIENT.

Answer is B.

bumpbot · Answer

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P is a point (not shown in the figure) in the plane containing O, R, a

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