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A certain circle in the xy-plane has its center at the origin. If P is

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A certain circle in the xy-plane has its center at the origin. If P is a point on the circle, what is the sum of the squares of the coordinates of P?

1) The radius of the circle is 4.

2) The sum of the coordinates of P is 0.
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Re: A certain circle in the xy-plane has its center at the origin. If P is [#permalink]

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Could someone please elaborate what the term "sum of the squares" mean in the context of coordinate geometry?
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A certain circle in the xy-plane has its center at the origin. If P is [#permalink]

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Hi Sallyzodiac,

The center-radius form of the circle equation is in the format \((x – h)^2\) + \((y – k)^2\) = \(r^2\), with the center being at the point (h, k) and the radius being "r". Now if we are told that the circle lies on the origin, this equation is reduced to \(x^2 + y^2 = r ^2\).

So, for us to calculate the squares of the coordinates - we just need to know the radius of the circle. Hence A.
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Re: A certain circle in the xy-plane has its center at the origin. If P is [#permalink]

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Sallyzodiac wrote:
A certain circle in the xy-plane has its center at the origin. If P is a point on the circle, what is the sum of the squares of the coordinates of P?

1) The radius of the circle is 4.

2) The sum of the coordinates of P is 0.


Say the coordinates of P are (x,y), then the question asks about the value of x^2 + y^2.

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


Image

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:
\(x^2+y^2=r^2\).

According to the above, the first statement of the question gives the direct answer: x^2 + y^2 = r^2 = 4^2. The second statement is not sufficient.

Check more here: math-coordinate-geometry-87652.html

Hope it helps.
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Re: A certain circle in the xy-plane has its center at the origin. If P is [#permalink]

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Hi Bunuel - My confusion is on the wording of the question- " If p is a point on the circle..." this is in fact saying that p has to be on the "outlined" circle and NOT for example inside the circle? If we choose p to be in the origin, 0,0 we would have another answer. But from what I am reading no one is considering P to be "inside" the circle. Are there any key clue to rule out points inside the circle?
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Bunuel wrote:
Sallyzodiac wrote:
A certain circle in the xy-plane has its center at the origin. If P is a point on the circle, what is the sum of the squares of the coordinates of P?

1) The radius of the circle is 4.

2) The sum of the coordinates of P is 0.


Say the coordinates of P are (x,y), then the question asks about the value of x^2 + y^2.

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


Image

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:
\(x^2+y^2=r^2\).

According to the above, the first statement of the question gives the direct answer: x^2 + y^2 = r^2 = 4^2. The second statement is not sufficient.

Check more here: math-coordinate-geometry-87652.html

Hope it helps.
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Re: A certain circle in the xy-plane has its center at the origin. If P is [#permalink]

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oloman wrote:
Hi Bunuel - My confusion is on the wording of the question- " If p is a point on the circle..." this is in fact saying that p has to be on the "outlined" circle and NOT for example inside the circle? If we choose p to be in the origin, 0,0 we would have another answer. But from what I am reading no one is considering P to be "inside" the circle. Are there any key clue to rule out points inside the circle?
best
Oloman


Bunuel wrote:
Sallyzodiac wrote:
A certain circle in the xy-plane has its center at the origin. If P is a point on the circle, what is the sum of the squares of the coordinates of P?

1) The radius of the circle is 4.

2) The sum of the coordinates of P is 0.


Say the coordinates of P are (x,y), then the question asks about the value of x^2 + y^2.

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


Image

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:
\(x^2+y^2=r^2\).

According to the above, the first statement of the question gives the direct answer: x^2 + y^2 = r^2 = 4^2. The second statement is not sufficient.

Check more here: math-coordinate-geometry-87652.html

Hope it helps.


Yes, on the circle means on the circumference, while in the circle means inside the circumference.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: A certain circle in the xy-plane has its center at the origin [#permalink]

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New post 27 Aug 2016, 15:17
Question asked here is, nothing the square of the radius of the circle, since the circle is centered at origin. Statement that helps in finding out the radius of the circle is sufficient.

Hence A.
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Re: A certain circle in the xy-plane has its center at the origin [#permalink]

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Keats wrote:
A certain circle in the xy-plane has its center at the origin. If P is a point on the circle, what is the sum of the squares of the coordinates of P?

(1)The radius of the circle is 4.
(2)The sum of the coordinates of P is 0.


We are given the center at origin, so equation of circle will be \(x^2 + y^2 = r^2\)

Since, P is on the circle, it will satisfy this equation.

Statement 1 : We are given the value of r = 4, so we can find out the value of \(x^2 + y^2\), Hence Sufficient.

Statement 2 says x = -y. Even if we put this value in the equation we will get \(2x^2 = r^2\)

But Since we don't know r, we cannot find the value of x. hence, insufficient.

Answer A.
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Re: A certain circle in the xy-plane has its center at the origin. If P is [#permalink]

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Re: A certain circle in the xy-plane has its center at the origin. If P is   [#permalink] 08 Sep 2017, 04:41
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