The circumcenter of the triangle formed by $(- 2, 3) (2, - 1)$ and $(4, 0)$ is

(3 / 2, 5 / 2)

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(- 3 / 2, 5 / 2)

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(3 / 2, - 5 / 2)

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(- 3 / 2, - 5 / 2)

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Solution

The correct option is A $(3 / 2, 5 / 2)$
$A = (2, 3), B = (2, - 1), C = (4, 0)$
$S (x, y)$ circumcenter
$S A^{2} = S B^{2}$
$\Rightarrow (x + 2)^{2} + (y - 3)^{2} = (y - 2)^{2} + (y + 1)^{2}$
$\Rightarrow x^{2} + 4 + 4 x + y^{2} + 9 - 6 y = x^{2} + 4 - 4 y + y^{2} + 1 + 2 y$
$\Rightarrow 8 x - 8 y + 8 = 0$
$\Rightarrow x - y + 1 = 0$ ........ $(1)$
$S B^{2} = S C^{2}$
$(x - 2)^{2} + (y + 1)^{2} = (x - 4)^{2} + (y - 0)^{2}$
$4 x + 2 y - 11 = 0$ ...... $(2)$
Circumcentre $(\frac{3}{2}, \frac{5}{2})$

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I. The centroid of the triangle formed by $(2, 3, - 1), (5, 6, 3), (2, - 3, 1)$ is	a) $(2, 2, 2)$
II. The circumcentre of the triangle formed by $(1, 2, 3), (2, 3, 1), (3, 1, 2)$ is	b) $(3, 1, 4)$
III. The orthocenter of the triangle formed by $(2, 1, 5), (3, 2, 3), (4, 0, 4)$ is	c) $(1, 1, 0)$
IV. The incentre of the triangle formed by $(0, 0, 0), (3, 0, 0), (0, 4, 0)$ is	d) $(3, 2, 1)$
	e) $(0, 0, 0)$

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