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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   45% (medium)

Question Stats: 58% (01:33) correct 42% (01:46) wrong based on 365 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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Math Expert V
Joined: 02 Sep 2009
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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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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.. ##### General Discussion
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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?
Math Expert V
Joined: 02 Sep 2009
Posts: 64242

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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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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. Math Expert V
Joined: 02 Sep 2009
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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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Can this be solved visually??
1) as a circle on the xy plane
OR
2) as a tringle(pitagoras)
Manager  Joined: 21 Feb 2012
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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.
>> !!!

You do not have the required permissions to view the files attached to this post.

Intern  B
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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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Joined: 18 Sep 2014
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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.
Intern  S
Joined: 30 May 2013
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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.
Manager  B
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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.
Intern  B
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Hi Brunel,

How are we suppose to verify answer E?

Cheers,
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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?
Intern  B
Joined: 07 Jul 2018
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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?
Math Expert V
Joined: 02 Sep 2009
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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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# D01-44

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