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If a circle with center at the origin passes through the point (0,5),

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If a circle with center at the origin passes through the point (0,5),  [#permalink]

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New post 16 Dec 2017, 02:08
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If a circle with center at the origin passes through the point (0,5), through which of the following points does it also pass?

(A) (2, √21)
(B) (4, 5)
(C) (5, √21)
(D) (5, 5)
(E) (5, 12)

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Re: If a circle with center at the origin passes through the point (0,5),  [#permalink]

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New post 16 Dec 2017, 03:58
The answer is A I suppose, coz the distance from origin to (0,5) is 5 so any option that satisfies the distance as 5 could be the answer. All suggestions accepted if I m wrong in good humour
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Karthik


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Re: If a circle with center at the origin passes through the point (0,5),  [#permalink]

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New post 16 Dec 2017, 06:52
circle with center at the origin passes through the point (0,5) , this means radius = 5.
we need to find another point which is at a distance of radius(5) from centre (0,0).
Only option A satisfies this...
Distance of (2, root 21) from (0,0) = (4+21)^0.5 = 5.

Hence Answer is A
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Re: If a circle with center at the origin passes through the point (0,5),  [#permalink]

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New post 16 Dec 2017, 08:33
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Bunuel wrote:
If a circle with center at the origin passes through the point (0,5), through which of the following points does it also pass?

(A) (2, √21)
(B) (4, 5)
(C) (5, √21)
(D) (5, 5)
(E) (5, 12)


Given: a circle centered at the origin is equal to coordinates (0,0) and the circle that passes through the point (0,5) has a radius equal to 5.
Question: find the choice that has the same distance as the radius of the circle = 5.

To calculate the distance between the origin and a point: \(\sqrt{(x^2+y^2)}\)

Note: most choices contain a 5 and anything with a square root greater than 25 is larger than 5, so eliminate all choices that contain 5 = BCDE.

(A) (2, √21) = \(\sqrt{(2^2+√21^2)} = √25 = 5\) this is the correct choice.

(B) (4, 5) = √(16+25) > 5
(C) (5, √21) = √(25+21) > 5
(D) (5, 5) = √(25+25) > 5
(E) (5, 12) = √(25+144) > 5
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If a circle with center at the origin passes through the point (0,5),  [#permalink]

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New post 19 Dec 2017, 09:43
Bunuel wrote:
If a circle with center at the origin passes through the point (0,5), through which of the following points does it also pass?

(A) (2, √21)
(B) (4, 5)
(C) (5, √21)
(D) (5, 5)
(E) (5, 12)

The basic equation of a circle yields the answer quickly.

A circle with center at origin* has the basic equation (Pythagorean theorem, where r = h):
\(x^2 + y^2 = r^2\)

Radius, r = 5. From (0,5) and (0,0): (5-0) = 5

This circle's equation is
\(x^2 + y^2 = 5^2\)
\(x^2 + y^2 = 25\)

Any two points on the circle must satisfy the equation: Square each coordinate. They must now sum to 25.

Eliminate answers B, C, D, and E

All have 5 as one coordinate.\(5^2 = 25\)
The other coordinate would have to be 0 for the Pythagorean theorem to hold.

By POE, the answer is A (2, √21)
And \((2^2 + √21^2) = (4 + 21) = 25\)

In case there is doubt:
\(x^2 + y^2 = 25\)

(B) (4, 5)
\((4^2 + 5^2) = (16 + 25) = 41\)
(C) (5, √21)
\((5^2 + √21^2) = (25 + 21) = 46\)
(D) (5, 5)
\((5^2 + 5^2) = (25 + 25) = 50\)
(E) (5, 12)
\((5^2 + 12^2) = (25 + 144) = 169\)

Answer A

*If the circle is not centered at origin, that is, has center (h,k), the general equation for a circle is used
\((x - h)^2 + (y - k)^2 = r^2\)

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If a circle with center at the origin passes through the point (0,5), &nbs [#permalink] 19 Dec 2017, 09:43
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