is the point (m, n) farther

[sanju09]

In the rectangular coordinate system, is the point (m, n) farther from the origin than point (p, q)?
(1) mn − pq = 12
(2) p + q = 21.

[Anurag@Gurome]

In the rectangular coordinate system, is the point (m, n) farther from the origin than point (p, q)?
(1) mn − pq = 12
(2) p + q = 21.

(1) mn − pq = 12
If p = 10, q = 0, m = 4, n = 3, then distance between (4, 3) and (0, 0) = √(4² + 3²) = 5 and distance between (10, 0) and (0, 0) = √(10² + 0²) = 10. Here, point (p, q) is farther from the origin than point (m, n).
If p = 1, q = 1, m = 1, n = 13, then distance between (1, 13) and (0, 0) = √(1² + 13²) = √170 and distance between (1, 1) and (0, 0) = √(1² + 1²) = √2. Here, point (m, n) is farther from the origin than point (p, q).
No definite answer; NOT sufficient.

(2) p + q = 21
If p = 21, q = 0, m = 4, n = 3, then distance between (4, 3) and (0, 0) = √(4² + 3²) = 5 and distance between (21, 0) and (0, 0) = √(21² + 0²) = 21. Here, point (p, q) is farther from the origin than point (m, n).
If p = 18, q = 3, m = 33, n = 2, then distance between (33, 2) and (0, 0) = √(33² + 2²) = √1093 and distance between (18, 3) and (0, 0) = √(18² + 3²) = √333. Here, point (m, n) is farther from the origin than point (p, q).
No definite answer; NOT sufficient.

Combining (1) and (2), we can take the same examples as in statement 2; again NOT sufficient.

The correct answer is E.

[krusta80]

In the rectangular coordinate system, is the point (m, n) farther from the origin than point (p, q)?
(1) mn − pq = 12
(2) p + q = 21.

Distance of (m,n) to the origin = sqrt(m^2+n^2)
Distance of (p,q) to the origin = sqrt(p^2+q^2)

Rephrasing the question: sqrt(m^2+n^2) > sqrt(p^2+q^2)?

It looks like this is yet another well-hidden application of FOIL. Let's yet again re-write the question.

sqrt[(m+n)^2 - 2mn] > sqrt[(p+q)^2 - 2pq]?

From (1) we have mn - pq = 12

mn = pq+12

Let's plug-in values for mn and pq -> nm = 13 and pq = 1

sqrt[(m+n)^2 - 26] > sqrt[(p+q)^2 - 1]?

Using values for m and n that are far apart (multiplicative inverses) and values for p and q that are equal makes this true, while the opposite makes it false. Note that there is no limit to how big we can make any one variable, since there is always an inverse value that can set mn or pq equal to 13 or 1, respectfully.


From Part (2) we have p+q = 21

Taken alone, this tells us nothing about m and n, so it must be INSUFFICIENT

(1) and (2) Together

This may be enough to solve now...

sqrt[(m+n)^2 - 2mn] > sqrt[(p+q)^2 - 2pq]?

sqrt[(m+n)^2 - 2(pq+12)] > sqrt(21^2 - 2pq)

sqrt[(m+n)^2 - 2pq - 24)] > sqrt(21^2 - 2pq)

Squaring both sides...

(m+n)^2 - 2pq - 24 > 21^2 - 2pq?
(m+n)^2 > 21^2 + 24?

We still have two unknowns here, so INSUFFICIENT

E