Question

The tangent to a circle of radius 8 cm from an external point P is of length 6 cm. Find the distance of P from the nearest point to the circle.

Answer 1

Given

Radius of circle = 8 cm.

Length of tangent from point P = 6 cm.

PA is tangent, O is centre, A is point of tangency.

Need to find: the distance of P from the nearest point on the circle.

That nearest point is the point where the line PO meets the circle.

Step 1: Use the tangent–radius right-angle property.

Radius to the point of tangency is perpendicular to the tangent:

OA ⊥ PA

Thus, triangle POA is right-angled at A.

Step 2: Apply Pythagoras

PO² = PA² + OA²

PO² = 6² + 8²

PO² = 36 + 64 = 100

PO = 10 cm

Step 3: Distance from P to the nearest point on the circle.

The nearest point lies on the line PO.
Distance = PO − radius:

Required distance = 10 − 8 = 2 cm