Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a
(A) Statement (1) alone is sufficient, but statement(2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement(1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, butNEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOTsufficient.
Answer - A
Statement (1) alone is not sufficient to answer the question as - depends upon sign of x and y.
Statement (2) alone is not sufficient to answer the question as - depends upon sign of x and y.
Together, statement (1) and (2) ---
On solving (1)
(x+y)^2 = 9a
x+y = + - 3 (a^1/2)
For x^2 + y^2 to be minimum x = y = + - 1.5 (a^1/2)
x^2 + y^2 = 2.25 a + 2.25 a
=4.5a > 4a
Thus answer is A
However if
a=0 then x=y=0 also, so this inequality will not hold true...
In that case it is very easy for a=x=y=0 ,no way x^2 + y^2 > 4a can be satisfied for any condition.
So answer will straight away be E.
This is assumption that x,y and a are different numbers.
Thus answer will be E in this case.
Official Answer to the above problem.
(1) INSUFFICIENT: If we multiply this equation out, we get:
x2 + 2xy + y2 = 9a
If we try to solve this expression for
x2 + y2, we getx2 + y2 = 9a – 2xy
Since the value of this expression depends on the value of x and y, we don't have enough information.
(2) INSUFFICIENT: If we multiply this equation out, we get:
x2 – 2xy + y2 = a
If we try to solve this expression for x2 + y2,
we getx2 + y2 = a + 2xy
Since the value of this expression depends on the value of x and y, we don't have enough information.
(1) AND (2) INSUFFICIENT: We can combine the two expanded forms of the equations from the two statements by adding them:
x2 + 2xy + y2 = 9ax2 – 2xy + y2 = a----- 2x2 + 2y2 = 10ax2 + y2 = 5a
If we substitute this back into the original question, the question becomes: "Is 5a > 4a?"If a > 0, the answer is yes.We know from the question stem that a is nonnegative.However, if a = 0 the answer is no.
The correct answer is E.
