The distance between the centres of two circles is 61 cm and their radii are 35 cm and 24 cm respectively. What is the length, in centimetres, of their direct common tangent?

Introduction / Context:
This problem concerns common tangents to two circles. When two circles have some distance between their centers, we can draw direct common tangents that touch both circles externally. Geometry provides a formula to find the length of such a tangent in terms of the distance between centers and the radii of the circles. This formula is very useful in competitive exams.

Given Data / Assumptions:

• The distance between the centers of the two circles is d = 61 cm.
• The radii of the circles are r1 = 35 cm and r2 = 24 cm.
• We must find the length of a direct common tangent between the two circles.
• The circles do not intersect and are in a standard Euclidean plane.

Concept / Approach:
For two non intersecting circles with distance between centers d and radii r1 and r2, the length of a direct common tangent L is given by: L = √(d^2 − (r1 − r2)^2) This formula comes from examining a right triangle formed by joining the centers and the points of tangency and applying the Pythagoras theorem. The difference of the radii appears because the direct tangent touches both circles externally on the same side.

Step-by-Step Solution:
Step 1: Identify the values in the formula. d = 61 cm, r1 = 35 cm, r2 = 24 cm. Step 2: Compute the difference of the radii. r1 − r2 = 35 − 24 = 11 cm. Step 3: Substitute into the formula for the direct common tangent length. L = √(d^2 − (r1 − r2)^2) L = √(61^2 − 11^2). Step 4: Compute the squares. 61^2 = 3721 11^2 = 121 Step 5: Subtract inside the square root. d^2 − (r1 − r2)^2 = 3721 − 121 = 3600 Step 6: Take the square root of 3600. L = √3600 = 60 cm.

Verification / Alternative check:
We can think of a right triangle whose hypotenuse is the line joining the centers (length 61 cm), one leg is the difference of the radii (11 cm), and the other leg is the length of the direct tangent between the points of contact. Applying Pythagoras theorem: 61^2 = 11^2 + L^2 ⇒ L^2 = 3721 − 121 = 3600 ⇒ L = 60. This matches the value obtained directly from the formula, confirming the answer.

Why Other Options Are Wrong:
Option 2: 54 cm would require L^2 = 2916, which is not equal to 3600, so it does not satisfy the right triangle relation. Option 3: 48 cm gives L^2 = 2304, far from the required 3600. Option 4: 72 cm gives L^2 = 5184, implying a much longer tangent than allowed by the geometry. Option 5: None of these is incorrect because 60 cm is listed and matches the exact calculation.

Common Pitfalls:
Students sometimes mistakenly use the sum of the radii instead of the difference when computing the direct common tangent. The sum is used when dealing with transverse tangents or other configurations. Another frequent error is miscomputing the squares or the subtraction (for example, mixing up 61^2). Also, some learners may attempt to apply a more complicated method involving coordinates and angles when the direct formula is much faster and less error prone. Remembering and correctly applying the tangent length formula avoids these mistakes.

Final Answer:
The length of the direct common tangent is 60 cm.