If the tangents P Q and P R are drawn to the circle x^2+...

If the tangents $P Q$ and $P R$ are drawn to the circle $x^{2}+$ $y^{2}=a^{2}$ from the point $P\left(x_{1}, y_{1}\right)$, then the equation of the circumcircle of $\triangle P Q R$ is (A) $x^{2}+y^{2}-x x_{1}-y y_{1}=0$ (B) $x^{2}+y^{2}+x x_{1}+y y_{1}=0$ (C) $x^{2}+y^{2}-2 x x_{1}-2 y y_{1}=0$ (D) none of these

If the tangents $P Q$ and $P R$ are drawn to the circle $x^{2}+$ $y^{2}=a^{2}$ from the point $P\left(x_{1}, y_{1}\right)$, then the equation of the circumcircle of $\triangle P Q R$ is
(A) $x^{2}+y^{2}-x x_{1}-y y_{1}=0$
(B) $x^{2}+y^{2}+x x_{1}+y y_{1}=0$
(C) $x^{2}+y^{2}-2 x x_{1}-2 y y_{1}=0$
(D) none of these

Key Concepts

Circle Geometry

Circle geometry is the study of the properties and relations of figures related to a circle, such as chords, tangents, and arcs. It includes understanding how lines and curves interact with the circle, which is crucial for analyzing problems involving tangents drawn from an external point and related constructions.

Tangents from an External Point

When two tangents are drawn from an external point to a circle, they are equal in length and subtend equal angles at the point of tangency. This concept is fundamental in solving problems related to the geometry of circles, as it introduces relationships between the tangents, the point, and the circle's radius.

Chord of Contact

The chord of contact is the line segment joining the points at which the two tangents from an external point touch the circle. This concept is important when studying the loci of points and angles formed in circle problems, as it provides a connection between the external point and the circle.

Circumcircle of a Triangle

The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. The center of this circle, called the circumcenter, is found at the intersection of the perpendicular bisectors of the triangle's sides. Understanding the circumcircle is key in various geometric constructions and proofs involving triangles and circles.

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