Question

If the shortest distance of a given point to a given circle is 4 cm and the longest distance is 9 cm, then the radius of the circle is:

• A. 2.5 cm or 6.5 cm
• B. 6.5 cm
• C. 5 cm or 13 cm
• D. 2.5 cm

Hint:

For problems involving the shortest and longest distances to a circle, use the relationships \( d - r \) and \( d + r \) to calculate the radius.

Answer 1

The Correct Option is A

Step 1: Use the formula for the shortest and longest distances to the circle. Let the radius of the circle be \( r \), and the distance from the point to the center of the circle be \( d \). The shortest distance from the point to the circle is \( d - r = 4 \) cm, and the longest distance is \( d + r = 9 \) cm. Solving these two equations: \[ d - r = 4 \quad \text{and} \quad d + r = 9. \] Adding the two equations gives: \[ 2d = 13 \quad \Rightarrow \quad d = 6.5 \, \text{cm}. \] Substituting \( d = 6.5 \) cm into \( d - r = 4 \) cm: \[ 6.5 - r = 4 \quad \Rightarrow \quad r = 2.5 \, \text{cm}. \] Thus, the radius of the circle is either \( 2.5 \) cm or \( 6.5 \) cm, depending on whether the point lies inside or outside the circle. Thus, the correct answer is (A).