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A circle passes through the point (5,0). What is the area of this

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A circle passes through the point (5,0). What is the area of this  [#permalink]

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14 Mar 2018, 00:00
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55% (hard)

Question Stats:

53% (01:20) correct 47% (01:37) wrong based on 52 sessions

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A circle passes through the point (5,0). What is the area of this circle?

(1) This circle also passes through points (3,4) and (-4,3).

(2) Centre of this circle is at (0,0).
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Re: A circle passes through the point (5,0). What is the area of this  [#permalink]

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14 Mar 2018, 13:57
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A circle passes through the point (5,0). What is the area of this circle?

(1) This circle also passes through points (3,4) and (-4,3).

The general form of the circle is x^2 + y^2 + Dx + Ey + F = 0

(3,4): 3^(2)+4^(2) + 3D + 4E + F =0 -> 3D + 4E + F = -25 * 3 -> 9D + 12E + 3F = -75 -> (1)

(4,-3): 4^(2)+(-3)^(2) + 4D - 3E + F =0 -> 4D - 3E + F = -25 * 4 -> 12D - 12E + 4F = -100 -> (2)

(5,0): (5)^(2)+5D+F = 0 -> 5D + F = -25 -> (3)

Adding equations (1) and (2), we get 21D + 7F = - 175 -> 3D + F = -25 -> (4)

Solving equations (3) and (4), we get D = 0 and F = -25

Substituting these values in (2), we get E = 0.

We will get the equation of the circle $$x^2 + y^2 = (5)^2$$

The area of this circle is $$25*\pi$$ (Sufficient)

(2) Centre of this circle is at (0,0).

If the circle's centre is (0,0) and it passes through the point (5,0)
The radius of the circle is the distance between these points, 5.

The area of this circle is $$25*\pi$$ (Sufficient - Option D)

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Re: A circle passes through the point (5,0). What is the area of this  [#permalink]

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14 Mar 2018, 16:20
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Given [5,0] lies on the circle.

(1) This circle also passes through points (3,4) and (-4,3).

let's say co-ordinate (x, y) denotes the centre of the circle .. distance from the centre to each of the given points will denote radius hence the distance will be same.

$$(x -5)^2$$+ $$y^2$$ = $$(x -3)^2$$+ $$(y-4)^2$$ = $$(x+4)^2$$+ $$(y-3)^2$$

we know $$3^2$$+ $$4^2$$ = 25 = $$5^2$$

so we make x = y = 0 in our equation : we get : 25 = 25 = 25 or 5 is the radius of the circle hence we can find the area.

(2) Centre of this circle is at (0,0). - radius = 5 hence we can find the area

ans : D
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Re: A circle passes through the point (5,0). What is the area of this  [#permalink]

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24 Mar 2018, 07:45
There is no need to calculate anything in this question.

1. Only one distinct circle can go through 3 points if they are not collinear. Sufficient
2. A point and a center is enough to construct the circle. Sufficient
Re: A circle passes through the point (5,0). What is the area of this   [#permalink] 24 Mar 2018, 07:45
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