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Math Expert V
Joined: 02 Sep 2009
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15 00:00

Difficulty:   45% (medium)

Question Stats: 60% (01:37) correct 40% (01:47) wrong based on 240 sessions

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$$x^2 + y^2 = 100$$. All of the following could be true EXCEPT

A. $$|x| + |y| = 10$$
B. $$|x| \gt |y|$$
C. $$|x| \gt |y| + 10$$
D. $$|x| = |y|$$
E. $$|x| - |y| = 5$$

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According to the question the maximum value of either x^2 or y^2 can be 100, which implies that the greatest absolute value of either x or y can be 10

option C indicates that absolute value of x is greater than 10 which cannot be the case.

kudos.. Intern  Joined: 25 Mar 2014
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Can this be solved visually??
1) as a circle on the xy plane
OR
2) as a tringle(pitagoras)
Math Expert V
Joined: 02 Sep 2009
Posts: 59075

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3
Official Solution:

$$x^2 + y^2 = 100$$. All of the following could be true EXCEPT

A. $$|x| + |y| = 10$$
B. $$|x| \gt |y|$$
C. $$|x| \gt |y| + 10$$
D. $$|x| = |y|$$
E. $$|x| - |y| = 5$$

A. $$|x| + |y| = 10$$ is possible if one is 0 and the other is 10.

B. $$|x| \gt |y|$$ is possible if $$|x| \gt |5\sqrt{2}|$$ and $$|y| \lt |5\sqrt{2}|$$

C. $$|x| \gt |y| + 10$$ is never possible because if $$|x| \gt 10$$, $$x^2+y^2$$ becomes greater than 100, which is wrong.

D. $$|x| = |y|$$ is possible if each is equal to $$|5\sqrt{2}|$$.

E. $$|x| - |y| = 5$$ is possible if $$|x| = |9.11|$$ and $$|y| = |4.11|$$.

Therefore all but C are possible. $$|x| \gt |y| + 10$$ means $$x$$ is greater than 10, which is not possible.

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Math Expert V
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Samwong wrote:
Bunuel wrote:
shankar245 wrote:
Hi Bunuel,

But the option E how is it possible as for the give n values the value of x^2 + y^2 =100 will not hold true correct?

Those are approximate values. Exact values are:
$$x = \frac{5}{2} (1+\sqrt{7})$$, $$y = \frac{5}{2} (\sqrt{7}-1)$$

Bunuel, can you please explain how you derive the exact values for x and y? Thanks.

30,000 Kudos. Awesome milestone. It really does not matter.
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plaverbach wrote:
Can this be solved visually??
1) as a circle on the xy plane
OR
2) as a tringle(pitagoras)

yes it can be solved by co-ordinate geometry.

x^2 + y^2 = 100
is a circle with radius 10

A. |x| + |y|=10 are four lines which intersect the circle at (10,0),(0,10),(-10,0)&(0,-10)
B. |x|>|y| there are multiple such points I have marked one such in the attached picture
C. This is an area outside the circle. The closest to the circle is at y=0 even here x is outside the area of the circle. A suggestion that when we look at inequalities in co-ordinate geometry think in terms of area.
D. |x|=|y| essentially two lines y=x and y=-x
E. |x| - |y|=5 they are four lines again

Attaching a very crude diagram. Apologies.
>> !!!

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JIAA wrote:
Bunuel wrote:
Official Solution:

$$x^2 + y^2 = 100$$. All of the following could be true EXCEPT

A. $$|x| + |y| = 10$$
B. $$|x| \gt |y|$$
C. $$|x| \gt |y| + 10$$
D. $$|x| = |y|$$
E. $$|x| - |y| = 5$$

A. $$|x| + |y| = 10$$ is possible if one is 0 and the other is 10.

B. $$|x| \gt |y|$$ is possible if $$|x| \gt |5\sqrt{2}|$$ and $$|y| \lt |5\sqrt{2}|$$

C. $$|x| \gt |y| + 10$$ is never possible because if $$|x| \gt 10$$, $$x^2+y^2$$ becomes greater than 100, which is wrong.

D. $$|x| = |y|$$ is possible if each is equal to $$|5\sqrt{2}|$$.

E. $$|x| - |y| = 5$$ is possible if $$|x| = |9.11|$$ and $$|y| = |4.11|$$.

Therefore all but C are possible. $$|x| \gt |y| + 10$$ means $$x$$ is greater than 10, which is not possible.

Xylan can you help?? What if i don't want to do number plugging??

JIAA It's completely okay if you do NOT want to do number plugging.
However, aspire to reach the CORRECT solution in the least possible time so that one can spend judicious time on 700+ Qs.

Quote:
Remember, Our arsenal should be equipped with all sorts of ammunition to tame the beast such as Reverse-solving, plugging different numbers, edge-case scenarios, etc.

The equation $$x^2 + y^2 = 100$$ is actually the locus of a circle with the origin as the center and radius of 10 units.
If possible Pre-Think the problem such as the allowable value of X and Y -
According to the question, the maximum value of either $$x^2$$ or $$y^2$$ can be 100, which implies that the greatest absolute value of either X or Y can be 10.
Thus, $$|x|$$ must be $$<= 10$$. Refer the attached picture.

Hence, $$|x|$$ CANNOT be $$> 10$$. Let alone $$|x|$$ being greater than $$|y| + 10$$.

If we take another look at the given statement $$x^2 + y^2 = 100$$, it can be considered an equation of right angle triangle with hypotenuse = 10 and perpendicular-sides as X and Y.
$$x^2 + y^2 = 100$$
And we know that the sum of two sides is always greater than the third side.
Thus: |y| + 10 > |x| : The third-side is smaller than the sum of other two-sides.
Therefore, OptionC is incorrect as it says $$|x| > |y| + 10$$, which can NEVER be true.

>> !!!

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Math Expert V
Joined: 02 Sep 2009
Posts: 59075

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shankar245 wrote:
Hi Bunuel,

But the option E how is it possible as for the give n values the value of x^2 + y^2 =100 will not hold true correct?

Those are approximate values. Exact values are:
$$x = \frac{5}{2} (1+\sqrt{7})$$, $$y = \frac{5}{2} (\sqrt{7}-1)$$
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Akshit03 wrote:
I am facing a concept problem in this question, can anyone help with it?

In option C: if x=$$\sqrt{65}$$ and $$y=\sqrt{35}$$

then x^2 + y^2 = 100 and
|x| > |y| + 35

because 65 > 35 + 10 ?

What concept have I got wrong?

So, if $$x = \sqrt{65}$$ and $$y=\sqrt{35}$$, then $$|x| =\sqrt{65} \approx 8.1$$ and $$|y|=\sqrt{35} \approx 6$$. Thus C is NOT true: 8.1 is not greater than 6 + 10 = 16.
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Hi Bunuel,

But the option E how is it possible as for the give n values the value of x^2 + y^2 =100 will not hold true correct?
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Bunuel wrote:
shankar245 wrote:
Hi Bunuel,

But the option E how is it possible as for the give n values the value of x^2 + y^2 =100 will not hold true correct?

Those are approximate values. Exact values are:
$$x = \frac{5}{2} (1+\sqrt{7})$$, $$y = \frac{5}{2} (\sqrt{7}-1)$$

Bunuel, can you please explain how you derive the exact values for x and y? Thanks.

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I think this the explanation isn't clear enough, please elaborate. please elaborate why C is right

and also why E is wrong?

for C : x^2 +y^2 -2xy>100 and we don't know anything about 2xy so how can we conclude C is right?
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mrigoel wrote:
I think this the explanation isn't clear enough, please elaborate. please elaborate why C is right
and also why E is wrong?
for C : x^2 +y^2 -2xy>100 and we don't know anything about 2xy so how can we conclude C is right?

Please note that this is an except question.
C cannot be right and that is why it is the answer.
E is wrong because it is possibly true for some exceptional values as indicated by Bunnel above.
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I thought of triangles and decided that C isn't possible although I think mapping the circle probably would have been quicker and less prone to error.
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1
I started with values of x & y as :
10 & 0, this rejects options (a) & (b)
5√2 & 5√2, this reject option (d)

For (c) & (e), if we take another look at the given statement, it can be consider an equation of right angle triangle.
x² + y² = 10²
And we know that sum of two sides is always greater than the third side.
Hence |x| > |y| + 10 can never be true.
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A, B and D can be quickly eliminated.

For A, we can have x=10, y=0. This makes B true as well immediately
For B, we can have x=y= 2root5

This leaves C and E. C looks easier to prove or disprove so start there. To maximise X, minimise Y. Abs val. Y cannot be negative so the smallest possible value for this is 0. That means the largest possible value for x is 10. X cannot be more than y+10 as y+10 = 10.

As we know this cannot be true, we don't even need to look at E
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this question can be thought of in a much easier manner for those who know a bit of coordinate geometry. the equation s actually the locus of a circle with origin as the centre and radius of 10 units.
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Hi Brunel,

How are we suppose to verify answer E?

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Using parametric notations, x = 10sin(t) and y=10cos(t) where t is a parameter. It is quite easy to point out C as the answer with this approach.
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Bunuel wrote:
shankar245 wrote:
Hi Bunuel,

But the option E how is it possible as for the give n values the value of x^2 + y^2 =100 will not hold true correct?

Those are approximate values. Exact values are:
$$x = \frac{5}{2} (1+\sqrt{7})$$, $$y = \frac{5}{2} (\sqrt{7}-1)$$

can someone kindly explain to me how this was derived? Re: D01-44   [#permalink] 18 Feb 2019, 05:47

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# D01-44

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