If a circle passes through the point (a, b) and cuts the...

If a circle passes through the point $(a, b)$ and cuts the circle $x^{2}+y^{2}=4$ orthogonally, then the locus of its centre is (A) $2 a x+2 b y+\left(a^{2}+b^{2}+4\right)=0$ (B) $2 a x+2 b y-\left(a^{2}+b^{2}+4\right)=0$ (C) $2 a x-2 b y+\left(a^{2}+b^{2}+4\right)=0$ (D) $2 a x-2 b y-\left(a^{2}+b^{2}+4\right)=0$

If a circle passes through the point $(a, b)$ and cuts the circle $x^{2}+y^{2}=4$ orthogonally, then the locus of its centre is
(A) $2 a x+2 b y+\left(a^{2}+b^{2}+4\right)=0$
(B) $2 a x+2 b y-\left(a^{2}+b^{2}+4\right)=0$
(C) $2 a x-2 b y+\left(a^{2}+b^{2}+4\right)=0$
(D) $2 a x-2 b y-\left(a^{2}+b^{2}+4\right)=0$

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Key Concepts

Orthogonality of Circles

In circle geometry, two circles are said to be orthogonal if at their intersections, the tangents to both circles are perpendicular. This condition leads to a relation between the centers and the radii of the circles, specifically that the square of the distance between the centers equals the sum of the squares of the radii. This concept is crucial when establishing relationships between circles that intersect at right angles.

Locus in Coordinate Geometry

The locus of a point is the set of all points that satisfy a specific condition or a set of conditions in the coordinate plane. In problems involving circles, deriving the locus of the center involves creating an equation that encapsulates the constraints imposed by given geometric conditions, such as passing through a fixed point or maintaining a particular geometric relationship with another circle.

Circle Passing Through a Fixed Point

When a circle passes through a given fixed point, the coordinates of that point satisfy the equation of the circle. This condition links the circle's center and radius, providing one of the necessary equations for determining unknown parameters in a circle's equation. It is often used in tandem with other conditions, such as orthogonality, to fully determine the circle.

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