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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