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 $x^{2}+y^{2}=p^{2}$ orthogonally, then the equation of the locus of its centre is |2005] (A) $x^{2}+y^{2}-3 a x-4 b y+\left(a^{2}+b^{2}-p^{2}\right)=0$ (B) $2 a x+2 b y-\left(a^{2}-\beta^{2}+p^{2}\right)=0$ (C) $x^{2}+y^{2}-2 a x-3 b y-\left(a^{2}-\beta^{2}-p^{2}\right)=0$ (D) $2 a x+2 b y-\left(a^{2}+b^{2}+p^{2}\right)=0$

If a circle passes through the point $(a, b)$ and cuts the circle $x^{2}+y^{2}=p^{2}$ orthogonally, then the equation of the locus of its centre is |2005]
(A) $x^{2}+y^{2}-3 a x-4 b y+\left(a^{2}+b^{2}-p^{2}\right)=0$
(B) $2 a x+2 b y-\left(a^{2}-\beta^{2}+p^{2}\right)=0$
(C) $x^{2}+y^{2}-2 a x-3 b y-\left(a^{2}-\beta^{2}-p^{2}\right)=0$
(D) $2 a x+2 b y-\left(a^{2}+b^{2}+p^{2}\right)=0$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Orthogonal Circles

Two circles are said to be orthogonal if they intersect at right angles, meaning the tangents at their intersection points are perpendicular. This condition leads to a specific relationship between the centers and radii of the two circles. Algebraically, if one circle has center (h1, k1) with radius r1 and the other (h2, k2) with radius r2, they are orthogonal if (h1 - h2)² + (k1 - k2)² = r1² + r2².

Locus of Points

A locus is the set of points that satisfy one or more given conditions. In geometry, determining the locus involves finding a common equation or relation that all these points adhere to. When applied to circle problems, for example, finding the locus of the centers of circles that satisfy certain constraints, such as passing through a fixed point and being orthogonal to another circle, results in an equation that describes this set of centers.

Circle Equation

A circle in the Cartesian plane is typically defined by the equation (x - h)² + (y - k)² = r², where (h, k) represents the center and r is the radius. This basic form allows us to derive and compare various properties of circles, such as intersections and tangents by analyzing how changes in h, k, and r affect the circle.

Transcript

Please give Ace some feedback