(A) The given points are $A(a \cos \alpha, a \sin \alpha)$,$B(a \cos \beta, a \sin \beta)$,and $C(a \cos \gamma, a \sin \gamma)$.These points lie on a

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(A) The given points are $A(a \cos \alpha, a \sin \alpha)$,$B(a \cos \beta, a \sin \beta)$,and $C(a \cos \gamma, a \sin \gamma)$.These points lie on a circle with center $(0, 0)$ and radius $a$,because the distance of each point from the origin $(0, 0)$ is $\sqrt{(a \cos \theta)^2 + (a \sin \theta)^2} = \sqrt{a^2(\cos^2 \theta + \sin^2 \theta)} = a$.Since all three vertices of the triangle lie on the circle centered at the origin,the circumcenter of the triangle is the origin $(0, 0)$.

Class 11 Mathematics 10-1.Circle and System of Circles Geometrical problems regarding circle and its properties

Difficult

Find the circumcenter of the triangle formed by the points $(a \cos \alpha, a \sin \alpha)$,$(a \cos \beta, a \sin \beta)$,and $(a \cos \gamma, a \sin \gamma)$.

A
$(0, 0)$
B
$\left( \frac{a}{3}(\cos \alpha + \cos \beta + \cos \gamma), \frac{a}{3}(\sin \alpha + \sin \beta + \sin \gamma) \right)$
C
$(a, 0)$
D
None of these

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10-1.Circle and System of Circles

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Geometrical problems regarding circle and its properties

If the points $(2, 0), (0, 1), (4, 5)$ and $(0, c)$ are concyclic,then $c$ is equal to

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If $P$ is a point on the circle $x^{2}+y^{2}=4$, $Q$ is a point on the straight line $5x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of the abscissae of all such points $P$ is ........... .

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Find the minimum and maximum distance from the point $(6, 8)$ to the circle $x^2 + y^2 = 4$.

Medium

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If the radical axis of the circles $x^2 + y^2 - 1 = 0$ and $x^2 + y^2 - 2x - 2y + 1 = 0$ forms a triangle of area $A$ with the coordinate axes,then the value of $\frac{1}{A}$ is

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If $5x - 12y + 10 = 0$ and $12y - 5x + 16 = 0$ are two tangents to a circle,then the radius of the circle is

EasyTS EAMCET 2003

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Find the circumcenter of the triangle formed by the points $(a \cos \alpha, a \sin \alpha)$,$(a \cos \beta, a \sin \beta)$,and $(a \cos \gamma, a \sin \gamma)$.

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