Multiple choice

The radii of two circles are $2$ units and $3$ units. lf the radical axis of the circles cuts one of the common tangents of the circle in $\mathrm{P}$ then ratio in which $\mathrm{P}$ divides the common tangent is

  1. $2 : 3$
  2. $3 : 2$
  3. $4 : 9$
  4. $1 : 1$
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D Correct answer
Explanation

For two circles, the radical axis is the locus of points from which the tangents to the two circles are equal in length. For any point P on the common tangent, the lengths of the tangents to both circles are equal. By the property of the radical axis and common tangents, the point P bisects the common tangent segment between the points of contact.

AI explanation

The radical axis is the locus of points with equal power with respect to both circles, so the lengths of the tangents drawn from point P to both circles are equal. If a line segment represents a common tangent touching the two circles at points A and B, then PA equals PB because both are tangent lengths from P to their respective circles. Since the distances from P to the two points of tangency are equal, P divides the common tangent in the ratio 1 to 1.

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