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 $x^{2}+y^{2}=$ $k^{2}$ orthogonally, then the equation of the locus of its centre is (a) $2 a x+2 b y-\left(a^{2}+b^{2}+k^{2}\right)=0$ (b) $2 a x+2 b y-\left(a^{2}-b^{2}+k^{2}\right)=0$ (c) $x^{2}+y^{2}-3 a x-4 b y+\left(a^{2}+b^{2}-k^{2}\right)=0$ (d) $x^{2}+y^{2}-2 a x-3 b y+\left(a^{2}-b^{2}-k^{2}\right)=0$

If a circle passes through the point $(a, b)$ and cuts the circle $x^{2}+y^{2}=$ $k^{2}$ orthogonally, then the equation of the locus of its centre is
(a) $2 a x+2 b y-\left(a^{2}+b^{2}+k^{2}\right)=0$
(b) $2 a x+2 b y-\left(a^{2}-b^{2}+k^{2}\right)=0$
(c) $x^{2}+y^{2}-3 a x-4 b y+\left(a^{2}+b^{2}-k^{2}\right)=0$
(d) $x^{2}+y^{2}-2 a x-3 b y+\left(a^{2}-b^{2}-k^{2}\right)=0$

Key Concepts

Circle Geometry

Circle geometry involves understanding the properties and equations of circles in a coordinate plane, including the standard form of a circle’s equation and the relationships between its radius and center. This is crucial for setting up problems involving circles passing through a specific point or interacting with other circles.

Orthogonal Circles

Two circles are said to be orthogonal if they intersect at right angles, which implies that at their points of intersection, the tangents to each circle are perpendicular. The algebraic condition for orthogonality between two circles links their radii and the distance between their centers, providing a key constraint in problems involving intersecting circles.

Locus in Coordinate Geometry

The concept of a locus in coordinate geometry refers to the set of all points that satisfy a given condition or a set of conditions. In problems involving circles and additional geometric constraints, determining the locus of a circle’s center helps to describe all possible circles that fulfill the stated properties.

Application of Geometric Conditions to Derive Equations

By applying the conditions that a circle must pass through a given point and be orthogonal to another circle, one can derive an equation that represents the locus of the center of the circle. This process involves substituting the conditions into the standard circle equation and utilizing algebraic manipulation to obtain a linear or other form of locus equation.

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