In the x-y coordinate plane, the distance between (p,q) and

This topic has expert replies
User avatar
Elite Legendary Member
Posts: 3991
Joined: Fri Jul 24, 2015 2:28 am
Location: Las Vegas, USA
Thanked: 19 times
Followed by:37 members

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

[Math Revolution GMAT math practice question]

In the x-y coordinate plane, the distance between (p,q) and (1,1) is 5. If p and q are integers, how many possibilities are there for the point (p,q)?

A. 2
B. 4
C. 8
D. 12
E. 16

GMAT/MBA Expert

Senior | Next Rank: 100 Posts
Posts: 38
Joined: Mon Mar 19, 2018 6:26 am
Followed by:1 members
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

In the x-y coordinate plane, the distance between (p,q) and (1,1) is 5. If p and q are integers, how many possibilities are there for the point (p,q)?

A. 2
B. 4
C. 8
D. 12
E. 16
To answer this question quickly, sketch a circle of radius 5, the center of which is located at point (1, 1).
On the circumference of the circle, the four points (1, 6), (1, -4), (-4, 1), and (6, 1) - which are above, below, left, and right of point (1, 1), respectively - are each 5 units from point (1, 1).

To quickly find the remaining points, consider a right triangle with a hypotenuse of 5, such that the triangle's hypotenuse coincides with the circle's radius.
If p and q are integers, then the legs of this right triangle have integer lengths. Only the 3-4-5 Pythagorean Triple satisfies this requirement, so the legs of this triangle have lengths of 3 and 4.

Consider the possible locations of point (p, q) in Quadrant I. Point (p, q) will be either 3 units to the right and 4 units above point (1, 1), or it will be 4 units to the right and 3 units above point (1, 1) -- points (4, 5) and (5, 4), respectively.
Similarly, there are two possible locations of point (p, q) in Quadrant II, two in Quadrant III, and two in Quadrant IV.

Thus, the total number of possible locations for point (p, q) is 12, and the correct answer is choice D.

User avatar
Elite Legendary Member
Posts: 3991
Joined: Fri Jul 24, 2015 2:28 am
Location: Las Vegas, USA
Thanked: 19 times
Followed by:37 members

by Max@Math Revolution » Mon Aug 13, 2018 5:12 am
=>
(p-1)^2 + (q-1)^2 = 5^2
If p - 1 = ±3, and q - 1 = ±4, then p = 1 ± 3, and q = 1 ± 4. There are four possible points: ( p, q ) = ( 4, 5 ), ( 4, -3 ), ( -2, 5 ), ( -2, -3 ).
If p - 1 = ±4, and q - 1 = ±3, then p = 1 ±4, and q = 1 ± 3. There are four possible points: ( p, q ) = ( 5, 4 ), ( 5, -2 ), ( -3, 4 ), ( -3, -2 ).
If p - 1 = 0, and q - 1 = ±5, then p = 1, and q = 1 ±5. There are two possible points: ( p, q ) = ( 1, 6 ), ( 1, -4 ).
If p - 1 = ±5, and q - 1 = 0, then p - 1 = ±5, and q = 1. There are two possible points: ( p, q ) = ( 6, 1 ), ( -4, 1 ).

There are a total of 4 + 4 + 2 + 2 = 12 possibilities for the point (p,q).
Therefore, the answer is D.

Answer: D