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# If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?

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Intern
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GMAT 1: 710 Q49 V39
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If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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27 Aug 2019, 04:05
2
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Difficulty:

95% (hard)

Question Stats:

21% (02:07) correct 79% (02:22) wrong based on 76 sessions

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If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions
Kudos are always appreciated
Director
Joined: 19 Oct 2018
Posts: 981
Location: India
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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27 Aug 2019, 06:49
2
1 Kudos for an excellent question.
IMO graphical approach can save lot of time in this question.

We just need to figure out whether point of minima of ||x|-0.34|+0.66=y lies within the circle or not, and whether local maxima at x=0 lies within circle or not.
$$x^2+y^2=1$$ is a curve of a circle of radius 1.

At x=0, ||x|-0.34|+0.66= 1, which lies on the circle.

Minima of $$||x|-0.34|+0.66=y$$ occurs when |x|-0.34=0 or x=+0.34 and -0.34, and y=0.66, lie inside the circle.

Now we are good to draw the curve.

There are 3 points where both curves meet.

D

Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions
Kudos are always appreciated

Attachments

1111.PNG [ 18.44 KiB | Viewed 1081 times ]

Intern
Status: Fortune favours the brave
Joined: 17 Jul 2019
Posts: 9
Location: Russian Federation
GMAT 1: 710 Q49 V39
GPA: 3.98
If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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27 Aug 2019, 06:52
3
Such tasks are easy to solve graphically: solutions to the system of equation are points of intersection of graphs of each equation.

Remember: When a module sign is "layed" on F(x), graphical part of F(x), which lays below the X-axis, is reflected over the X-axis. This operation gives you the graph of |F(x)|.

Thus, the given task is solved with consequent graphics:

As already discussed, to obtain ||x|-0.34| graph, we reflect "negative" part of |x|-0.34 over X-axis. Then shift ||x|-0.34| by 0.66 upwards to obtain ||x|-0.34|+0.66. Than we draw the plot of x^2+y^2=1

Finally, one can see 3 points of intersection -> 3 solutions to the system -> x can take 3 values -> answer is D
Intern
Joined: 22 Aug 2019
Posts: 1
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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27 Aug 2019, 07:38
This question is a tough one
Manager
Joined: 17 Jul 2017
Posts: 194
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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27 Aug 2019, 08:42
Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions
Kudos are always appreciated

nick1816
Bunuel
What is wrong with the given solution in the image:
here x=0 or 1 0r -1

but i consider case last one in pic (from d)
here i get x=.84 and y =-.52
then also x^2+y^2=1 then y are we nopt considering such cases
Attachments

IMG20190827210630.jpg [ 3.48 MiB | Viewed 1011 times ]

IMG20190827210624.jpg [ 3.52 MiB | Viewed 1014 times ]

Director
Joined: 19 Oct 2018
Posts: 981
Location: India
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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27 Aug 2019, 09:23
There are couple of problems with this approach for this particular question.
1. This approach can take a lot of time. Whenever you see second order equations, go for the graphical approach.

2. You didn't find the domain for x, for each equation.
For example-
||x|-0.34|+0.66=y
y=x-0.34+0.66, when x≥0.34
y=0.34-x+0.66, when $$0.34 ≥x ≥0$$
y=0.34+x+0.66, when 0≥x≥-0.34
y=0.34-x+0.66, when x≤-0.34

Now you can solve each case, and look whether the solution lies in the domain or not.

vanam52923 wrote:
Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions
Kudos are always appreciated

nick1816
Bunuel
What is wrong with the given solution in the image:
here x=0 or 1 0r -1

but i consider case last one in pic (from d)
here i get x=.84 and y =-.52
then also x^2+y^2=1 then y are we nopt considering such cases
Manager
Joined: 17 Jul 2017
Posts: 194
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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27 Aug 2019, 10:08
nick1816 wrote:
There are couple of problems with this approach for this particular question.
1. This approach can take a lot of time. Whenever you see second order equations, go for the graphical approach.

2. You didn't find the domain for x, for each equation.
For example-
||x|-0.34|+0.66=y
y=x-0.34+0.66, when x≥0.34
y=0.34-x+0.66, when $$0.34 ≥x ≥0$$
y=0.34+x+0.66, when 0≥x≥-0.34
y=0.34-x+0.66, when x≤-0.34

Now you can solve each case, and look whether the solution lies in the domain or not.

vanam52923 wrote:
Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions
Kudos are always appreciated

nick1816
Bunuel
What is wrong with the given solution in the image:
here x=0 or 1 0r -1

but i consider case last one in pic (from d)
here i get x=.84 and y =-.52
then also x^2+y^2=1 then y are we nopt considering such cases

But are such questions asked on GMAT?
SVP
Joined: 03 Jun 2019
Posts: 1724
Location: India
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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27 Aug 2019, 10:20
nick1816 VeritasKarishma Bunuel chondro48 Is there an algebra way to solve this question?

nick1816 wrote:
1 Kudos for an excellent question.
IMO graphical approach can save lot of time in this question.

We just need to figure out whether point of minima of ||x|-0.34|+0.66=y lies within the circle or not, and whether local maxima at x=0 lies within circle or not.
$$x^2+y^2=1$$ is a curve of a circle of radius 1.

At x=0, ||x|-0.34|+0.66= 1, which lies on the circle.

Minima of $$||x|-0.34|+0.66=y$$ occurs when |x|-0.34=0 or x=+0.34 and -0.34, and y=0.66, lie inside the circle.

Now we are good to draw the curve.

There are 3 points where both curves meet.

D

Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions
Kudos are always appreciated

_________________
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If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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Updated on: 28 Aug 2019, 00:39
Kinshook wrote:
nick1816 VeritasKarishma Bunuel chondro48 Is there an algebra way to solve this question?

nick1816 wrote:
1 Kudos for an excellent question.
IMO graphical approach can save lot of time in this question.

We just need to figure out whether point of minima of ||x|-0.34|+0.66=y lies within the circle or not, and whether local maxima at x=0 lies within circle or not.
$$x^2+y^2=1$$ is a curve of a circle of radius 1.

At x=0, ||x|-0.34|+0.66= 1, which lies on the circle.

Minima of $$||x|-0.34|+0.66=y$$ occurs when |x|-0.34=0 or x=+0.34 and -0.34, and y=0.66, lie inside the circle.

Now we are good to draw the curve.

There are 3 points where both curves meet.

D

Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

Kudos are always appreciated

Hi Kinshook nick1816 VeritasKarishma and Bunuel,

Yes, there is. Below is a framework to do so. Correct me, experts, if I make mistake.

Note that using algebra, the solution is laborious and time consuming.

Transform $$x^2=1-y^2$$ to |x|=√(1-y^2). Substituting |x| with √(1-y^2) into the second equation, we get:
|√(1-y^2)-0.34|=y-0.66.

If √(1-y^2) > 0.34, then √(1-y^2)-0.34 = y-0.66 --> √(1-y^2)= y-0.34. After squaring both sides and performing tedious calculation, we will find that there are two solution points available when √(1-y^2) > 0.34.

If √(1-y^2) <= 0.34, then 0.34-√(1-y^2) = y-0.66 --> √(1-y^2)=1-y. It is obvious that y=1 satisfies the equation and correspondingly |x|=√(1-y^2)=0. Thus, there is one solution point (0,1) when √(1-y^2) <= 0.34.

Finally, there are 3 values that x can take.

Originally posted by chondro48 on 27 Aug 2019, 17:04.
Last edited by chondro48 on 28 Aug 2019, 00:39, edited 1 time in total.
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Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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28 Aug 2019, 00:36
Kinshook wrote:
nick1816 VeritasKarishma Bunuel chondro48 Is there an algebra way to solve this question?

nick1816 wrote:
1 Kudos for an excellent question.
IMO graphical approach can save lot of time in this question.

We just need to figure out whether point of minima of ||x|-0.34|+0.66=y lies within the circle or not, and whether local maxima at x=0 lies within circle or not.
$$x^2+y^2=1$$ is a curve of a circle of radius 1.

At x=0, ||x|-0.34|+0.66= 1, which lies on the circle.

Minima of $$||x|-0.34|+0.66=y$$ occurs when |x|-0.34=0 or x=+0.34 and -0.34, and y=0.66, lie inside the circle.

Now we are good to draw the curve.

There are 3 points where both curves meet.

D

Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions
Kudos are always appreciated

You can solve it using the usual method of definition of absolute values (as done by others here) but if you are unable to solve it graphically, it is probably a good idea to skip the question. With algebra, it will be far too time consuming and error prone.
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Karishma
Veritas Prep GMAT Instructor

Intern
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Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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29 Aug 2019, 11:01
nick1816

The explanation given by you to find out the value of y at the equation's minima is crystal clear. We have arrived at y = 0.66 at minima i.e x = +/- 0.34.

Now, how do we figure if y = 0.66 falls inside the circle?

Ofcourse we can substitute x = 0.34 in the circle equation to find out the value of y. And if the value of y > 0.66, we can deduce that 0.66 is inside the circle. However, this is a time consuming process that requires squaring and square-roots

Is there are short cut to find if 0.66 falls inside the circle?

Director
Joined: 19 Oct 2018
Posts: 981
Location: India
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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29 Aug 2019, 11:12
1
0.34≃1/3
0.66≃2/3
$$(\frac{1}{3})^2+(\frac{2}{3})^2$$= $$\frac{1}{9}+\frac{4}{9}$$=$$\frac{5}{9}$$<1

GMATaspirant641 wrote:
nick1816

The explanation given by you to find out the value of y at the equation's minima is crystal clear. We have arrived at y = 0.66 at minima i.e x = +/- 0.34.

Now, how do we figure if y = 0.66 falls inside the circle?

Ofcourse we can substitute x = 0.34 in the circle equation to find out the value of y. And if the value of y > 0.66, we can deduce that 0.66 is inside the circle. However, this is a time consuming process that requires squaring and square-roots

Is there are short cut to find if 0.66 falls inside the circle?

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9706
Location: Pune, India
If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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29 Aug 2019, 23:13
GMATaspirant641 wrote:
nick1816

The explanation given by you to find out the value of y at the equation's minima is crystal clear. We have arrived at y = 0.66 at minima i.e x = +/- 0.34.

Now, how do we figure if y = 0.66 falls inside the circle?

Ofcourse we can substitute x = 0.34 in the circle equation to find out the value of y. And if the value of y > 0.66, we can deduce that 0.66 is inside the circle. However, this is a time consuming process that requires squaring and square-roots

Is there are short cut to find if 0.66 falls inside the circle?

Check the solution given by Philipp98 above. Notice how the graph of ||x| - 0.34| + 0.66 = y is drawn.
Also note that each green line has slope 1 or -1. Since the graph intersects the circle at (0, 1), the line of slope 1 will be inside the circle (45 degrees line). Hence you know that it will intersect the circle in 3 points.

Check out these two posts on how to draw absolute value graphs:

https://www.veritasprep.com/blog/2011/0 ... h-to-mods/
https://www.veritasprep.com/blog/2014/1 ... solutions/
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Karishma
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If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?   [#permalink] 29 Aug 2019, 23:13
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