(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

The number of circles having radius $5$ and passing through the points $(-2, 0)$ and $(4, 0)$ is

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In a $\triangle ABC$,a point $D$ is chosen on $BC$ such that $BD:DC = 2:5$. Let $P$ be a point on the circumcircle of $\triangle ABC$ such that $\angle PDB = \angle BAC$. Then $PD:PC$ is:

KVPY 2020

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The power of a point $(2, -1)$ with respect to a circle $C$ of radius $4$ is $9$. The centre of the circle $C$ lies on the line $x+y=0$ and in the $2^{\text{nd}}$ quadrant. If $(\alpha, \beta)$ is the centre of the circle $C$,then $\beta-\alpha=$

MediumTS EAMCET 2025

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If $P\left(\frac{7}{5}, \frac{6}{5}\right)$ is the inverse point of $A(1,2)$ with respect to a circle with centre $C(2,0)$,then the radius of that circle is

MediumTS EAMCET 2025

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$A$ point $P$ is taken outside $\Delta ABC$ where $B(1, \sqrt{3})$,$A(0, 0)$,and $C(2, 0)$,but inside the acute angle $BAC$,such that $\angle APC = \frac{\pi}{6}$ and $\angle BPA = \frac{\pi}{12}$. The slope of the line $BP$ is:

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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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