The locus of the mid-point of the chord of contact of...

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x-5 y=20$ to the circle $x^{2}+$ $y^{2}=9$ is (a) $20\left(x^{2}+y^{2}\right)-36 x+45 y=0$ (b) $20\left(x^{2}+y^{2}\right)+36 x-45 y=0$ (c) $36\left(x^{2}+y^{2}\right)-20 x+45 y=0$ (d) $36\left(x^{2}+y^{2}\right)+20 x-45 y=0$

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x-5 y=20$ to the circle $x^{2}+$ $y^{2}=9$ is
(a) $20\left(x^{2}+y^{2}\right)-36 x+45 y=0$
(b) $20\left(x^{2}+y^{2}\right)+36 x-45 y=0$
(c) $36\left(x^{2}+y^{2}\right)-20 x+45 y=0$
(d) $36\left(x^{2}+y^{2}\right)+20 x-45 y=0$

Key Concepts

Chord of Contact

When two tangents are drawn from an external point to a circle, the chord that connects the points at which the tangents touch the circle is called the chord of contact. Its equation can be derived using the tangent formulation of the circle, and it plays a crucial role in problems involving tangency and loci.

Pole and Polar Relationship

The pole-polar relationship is a fundamental concept in the geometry of conic sections. For a given circle, every point (the pole) outside the circle has an associated line (the polar) such that the chord of contact of the tangents from the pole to the circle lies on this line. This relationship helps in deriving conditions and equations involving tangents and chords.

Locus of a Point

A locus is a set of points that satisfy a particular condition or a set of conditions. In coordinate geometry, finding the locus involves eliminating parameters to obtain an equation that represents all possible positions of a point that meets those criteria, such as the midpoint of a chord of contact.

Tangents from an External Point to a Circle

From any point outside a circle, two tangents can be drawn. The equations of these tangents, and the conditions under which they exist (e.g., the tangency condition), are essential in solving problems that involve contact points, chords of contact, and related geometric constructions.

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