Two circles have radii 3 cm and 4 cm respectively, and the distance between their centres is 10 cm. What is the ratio of the length of a direct common tangent to the length of a transverse (internal) common tangent?

Introduction / Context:
This question involves geometric properties of tangents drawn to two circles. You are asked to compare the length of a direct common tangent with the length of a transverse (internal) common tangent. The problem checks whether you know the standard formulas relating the distance between centres and the radii of the circles to the lengths of these tangents.

Given Data / Assumptions:

• Radius of the first circle, r1 = 3 cm.
• Radius of the second circle, r2 = 4 cm.
• Distance between the centres of the circles, d = 10 cm.
• We must find the ratio (length of direct common tangent) : (length of transverse common tangent).
• Both circles are separate and the tangents considered are the usual external and internal common tangents.

Concept / Approach:
For two circles with radii r1 and r2 and centre distance d, there are formulas for the lengths of the common tangents:
Length of a direct (external) common tangent, L_d = √(d^2 - (r1 - r2)^2). Length of a transverse (internal) common tangent, L_t = √(d^2 - (r1 + r2)^2). We simply substitute the given values into these formulas and then form the ratio L_d : L_t, simplifying if possible.

Step-by-Step Solution:
Step 1: Identify given values: d = 10, r1 = 3, r2 = 4. Step 2: Compute r1 - r2 = 3 - 4 = -1; the square is (r1 - r2)^2 = (-1)^2 = 1. Step 3: Compute r1 + r2 = 3 + 4 = 7; the square is (r1 + r2)^2 = 7^2 = 49. Step 4: Find L_d: L_d = √(d^2 - (r1 - r2)^2) = √(10^2 - 1) = √(100 - 1) = √99. Step 5: Find L_t: L_t = √(d^2 - (r1 + r2)^2) = √(10^2 - 49) = √(100 - 49) = √51. Step 6: Ratio L_d : L_t = √99 : √51. Step 7: Simplify the ratio by factoring: √99 = √(9 * 11) = 3√11 and √51 = √(3 * 17) = √3 * √17. Step 8: Write both in terms of √3: √99 = √3 * √33, and √51 = √3 * √17. Step 9: Cancel the common factor √3, giving ratio = √33 : √17.

Verification / Alternative check:
You can also keep the ratio as √99 : √51 and compare it with the options. Noticing that 99 and 51 share a factor 3, divide both under the radical by 3 to get √(99 / 3) : √(51 / 3) = √33 : √17. This directly matches option √33 : √17, confirming that the algebraic simplification is correct without needing to evaluate any decimal approximations.

Why Other Options Are Wrong:
The option √51 : √68 would require the internal tangent to have length √68, which does not match the given formula and data. The ratio √66 : √51 has the two numbers reversed and does not follow from the calculations. The ratio √28 : √17 does not correspond to any combination of d^2, (r1 - r2)^2 or (r1 + r2)^2 for the given values. The option 1 : 1 would mean the direct and transverse tangents have equal length, which only occurs in special symmetric cases not satisfied here. Only √33 : √17 matches the correct ratio.

Common Pitfalls:
A common mistake is to swap the formulas and use r1 + r2 where r1 - r2 is needed, or vice versa. Another issue is forgetting to square the differences or sums before subtracting from d^2. Some students also simplify the radicals incorrectly, losing common factors and getting a ratio that does not match any option. Careful substitution and stepwise simplification of the radicals is essential for accuracy.

Final Answer:
The ratio of the length of the direct common tangent to the length of the transverse common tangent is √33 : √17.