In the xy-plane, point (r, s) lies on a circle with center at the orig

Question

In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of r^2 + s^2?

(1) The circle has radius 2
(2) The point  (\sqrt{2}, \ -\sqrt{2})  lies on the circle

DS02741

Bunuel · Accepted Answer

SOLUTION

THEORY:
In an xy-plane, the circle with center (a, b) and radius r is the set of all points (x, y) such that:
\((x-a)^2+(y-b)^2=r^2\)

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram above, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x-a and y-b.

If the circle is centered at the origin (0, 0), then the equation simplifies to: \(x^2+y^2=r^2\).

For more on this subject check Coordinate Geometry chapter of Math Book: https://gmatclub.com/forum/math-coordina ... 87652.html

BACK TO THE ORIGINAL QUESTION:
In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of \(r^2 + s^2\)?

Now, as \(x^2+y^2=r^2\) then the question asks about the value of radius^2.

(1) The circle has radius 2 --> radius^2=4. Sufficient.

(2) The point \((\sqrt{2}, \ -\sqrt{2})\) lies on the circle --> substitute x and y coordinates of a point in \(x^2+y^2=r^2\) --> \(2+2=4=r^2\). Sufficient.

Answer: D.

ConnectTheDots · Answer

It is the language.
On the circle = On the circumference.
In/Inside/Within the circle = Points enclosed by the circumference

ryusei1989 · Answer

Thanks for the brilliant explanation. One thing I don't get the question is that, the point (r,s) could be anywhere in the circle, not only on its circumference. Why does it refer only to a point on the circumference? Thanks!

nutshell · Answer

If a circle, lying on a xy-plane, has its center at the origin, the equation is x^2+y^2=R^2, where x & y are points on the circle and R is the radius of the circle.

Since x & y from the equation x^2+y^2=R^2 is similar to r & s in the question, we can rewrite as r^2+s^2=R^2.

Statement (1) gives us the value of radius, R; Therefore, r^2+s^2=4; Sufficient.
Statement (2) gives us the value of a point on the circle
=> sub x & y values of the point in r^2+s^2=R^22: 2+2 = 4; 
r^2 = 4; Sufficient.

Hence, Answer is (D).

noTh1ng · Answer

Hi,

i don't understand from  Bunuel s solution how we come up with the value of S in statement 1 in order to answer what r^2 + s^2 is?

r^2 equals 4, that is all clear, but s should be y-value, how do we get that?

Bunuel · Answer

r^2+s^2=radius^2 . (1) says that radius = 2, thus  r^2+s^2=2^2 .

GMATinsight · Answer

Hi  noTh1ng ,

In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of r^2 + s^2?

(1) The circle has radius 2
(2) The point (2√, −2√) lies on the circle

You seem to have misunderstood a little here.

The equation of Circle is given by  x^2 + y^2 = Radius^2

Given : (r,s) lie on the circle  
i.e. (r,s) will satisfy the equation of Circle
i.e.  r^2 + s^2 = Radius^2

Question : Find the value of  r^2 + s^2 ? but since  r^2 + s^2 = Radius^2  therefore, the question becomes

Question : Find the value of  Radius^2 ?

Statement 1 : The circle has radius 2 
i.e.  r^2 + s^2 = Radius^2 = 2^2 = 4 
 SUFFICIENT

Statement 2 : The point (√2, −√2) lies on the circle 
i.e. (√2, −√2) will satisfy the equation of circle 
i.e.  (√2)^2 + (−√2)^2 = Radius^2 
i.e. Radius = 4 
hence,  r^2 + s^2 = Radius^2 = 2^2 = 4  Hence,
 SUFFICIENT

Answer: Option D

I hope it helps!

Please Note: You have been confused r (X-co-ordinate) and r (Radius) as it seems from your question

WilDThiNg · Answer

Hi,

A very fundamental or, maybe, a silly question, in GMAT, when a question like this reads 'on a circle', it's supposed to mean, on the circumference of the circle, and not the whole of the area of the circle.

Thanks

Bunuel · Answer

Yes, on the circle means on the circumference. 
In the circle means within.

parkerd · Answer

So, for statement 2, any coordinates that are  on the circle  can be used to substitute for (r,s)?

Bunuel · Answer

Yes, if a circle is centred at the origin, then the x and y-coordinates of any point on the circle (so on the circumference) will satisfy  x^2+y^2=radius^2

parkerd · Answer

If the (r,s) was inside the circle would you still be able to solve with the same equation like in statement 1? Or be able to substitute for (r,s) with the given values on the circle like in statement 2?

Bunuel · Answer

If (r, s) were IN the circle, then the answer would be E because each statement gives us basically the same info - the length of the radius. How can we find the sum of the squares of a random point inside the circle just knowing the radius?

dabaobao · Answer

If a point (a, b) is on a circle with center at origin and radius = r, then a^2 + b^2 = r^2

Simply Q: r = ?

1) Sufficient
2) 2 + 2 = 4 = r^2 => Sufficient

ANSWER: D

gayatri259 · Answer

Statement 1 - r and s are two different points on a circle so both have values of 2^2 and 2^ so r^2+s^2 = 8
Is this the correct inference?

Basshead · Answer

That's not correct.

The question stem tells us the point lies on the circle with center at the  origin  (0,0). If the radius is 2, this means the coordinates could be (2,0), (0,2), etc.)

r^2 + s^2  will always equal 4.

ScottTargetTestPrep · Answer

Solution:

Question Stem Analysis:

We need to determine the value of r^2 + s^2, given that (r, s) is a point on the circle with center at the origin. Notice that the equation of such a circle is x^2 + y^2 = R^2 where R is the radius of the circle. Since (r, s) is a point on the circle, we have r^2 + s^2 = R^2. That is, in order to determine the value of r^2 + s^2, we either need to know the coordinates of a point on the circle or just the value of R.

Statement One Alone:
 
Since the radius of the circle is 2, R = 2. Therefore, r^2 + s^2 = 2^2 = 4. Statement one alone is sufficient.

Statement Two Alone:

Since (√2, -√2) is a point on the circle, x = √2 and y = -√2 will satisfy x^2 + y^2 = R^2. Substituting, we find R^2 = x^2 + y^2 = (√2)^2 + (-√2)^2 = 2+ 2 = 4. Proceeding as above, we obtain r^2 + s^2 = 4. Statement two alone is sufficient.

Answer: D

egmat · Answer

You're thinking that r and s are two different points on the circle. That's not correct!

(r,s) represents ONE point with coordinates :
 - r = x-coordinate
- s = y-coordinate

Circle Equation:  For ANY point (r,s) on a circle centered at the origin with radius R:
 r2 + s2 = R2

Statement 1 Analysis: 
The circle has radius  2 
Therefore, for ANY point (r,s) on this circle: r2 + s2 =  2 2 =  4

Examples of points on this circle: 
- Point ( 2 ,  0 ): r2 + s2 =  2 2 +  0 2 =  4  ✓
- Point ( 0 ,  2 ): r2 + s2 =  0 2 +  2 2 =  4  ✓
- Point ( √2 ,  √2 ): r2 + s2 =  (√2) 2 +  (√2) 2 =  2  +  2  =  4  ✓

Notice: ALL points on the circle give the same value for r2 + s2 
 
 Why your calculation r2 + s2 = 8 is wrong: 
You can't have both r =  2  AND s =  2  on a circle with radius  2 .
If r =  2  and s =  2 , then the distance from origin = √( 2 2 +  2 2) = √ 8  =  2√2  ≈  2.83 
This point would be OUTSIDE the circle!

Answer: D (Both statements alone are sufficient)

In the xy-plane, point (r, s) lies on a circle with center at the orig

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