Multiple choice

The chord of contact of the tangents drawn from a point on the circle, ${x^2} + {y^2} = {a^2}$ to the circle ${x^2} + {y^2} = {B^2}$ touch the circle ${x^2} + {y^2} = {c^2}$ then a,b,c are in :

  1. A.P.

  2. G.P.

  3. H.P.
  4. $None of these$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let a point on the circle x^2 + y^2 = a^2 be (a cos theta, a sin theta). The chord of contact of this point with respect to the circle x^2 + y^2 = b^2 is given by ax cos theta + ay sin theta = b^2. Since this line touches the circle x^2 + y^2 = c^2, its perpendicular distance from the origin must equal c, which gives b^2 / a = c, or b^2 = ac. This confirms that a, b, and c are in Geometric Progression (G.P.).

AI explanation

For a point (x1, y1) on the circle x^2 + y^2 = a^2, the equation of the chord of contact drawn to the circle x^2 + y^2 = b^2 is given by T = 0 as xx1 + yy1 = b^2. The perpendicular distance from the origin to this chord of contact line, which acts as the tangent to the circle x^2 + y^2 = c^2, must equal the radius c. Since x1^2 + y1^2 = a^2, the distance simplifies to b^2 / sqrt(x1^2 + y1^2) = b^2 / a. Setting this equal to c yields b^2 / a = c, which rearranges to b^2 = ac. This is the defining condition for a, b, and c to be in geometric progression (G.P.).

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