The locus of the centres of the circles which touch the two...

The locus of the centres of the circles which touch the two circles $x^{2}+y^{2}=a^{2}$ and $x^{2}+y^{2}=4 a x$ externally is
(A) $12 x^{2}-4 y^{2}-24 a x+9 a^{2}=0$
(B) $12 x^{2}+4 y^{2}-24 a x+9 a^{2}=0$
(C) $12 x^{2}-4 y^{2}+24 a x+9 a^{2}=0$
(D) none of these

Key Concepts

Standard Equation of a Circle

The equation of a circle in the Cartesian plane is typically written as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form is fundamental because it allows one to identify the center and radius directly, which is essential when analyzing conditions such as tangency between circles.

Tangency Condition for Circles

When two circles are externally tangent, the distance between their centers is equal to the sum of their radii. This condition is used to set up equations relating the center coordinates and radii of the circles involved, thereby forming the foundation for deriving the locus of centers for circles satisfying the tangency requirement.

Locus of Points

In geometry, the locus is the set of all points that satisfy a particular condition or a set of conditions. When the problem involves finding the centers of all circles that are tangent to given circles, one is essentially determining a locus, which often results in a conic section or another algebraic curve.

Conic Sections and Their Equations

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They include ellipses, parabolas, and hyperbolas. The equations derived from geometric conditions like tangency often simplify to quadratic forms that represent these conic sections. Identifying the type of conic section can provide insights into the geometric properties of the locus.

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