**All Squares --- Data sufficiency Question!**

**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.