The centers of two circles of radii 25 cm and 35 cm are 80 cm apart. What is the ratio of the lengths of the transverse common tangent to the direct common tangent to these circles?

Correct Answer: Option 4 ($2:3$)

Step-by-Step Explanation:

Step 1: Identify the given values

Let the radii of the two circles be $r_1$ and $r_2$, and the distance between their centers be $d$.

Given:

Radius of the first circle, $r_1=25\text{ cm}$

Radius of the second circle, $r_2=35\text{ cm}$

Distance between the centers, $d=80\text{ cm}$

Step 2: Calculate the length of the Transverse Common Tangent (TCT)

The formula for the length of a transverse common tangent is given by:

$\text{Length of TCT}=\sqrt{d^2-{(r_1+r_2)}^2}$

Substitute the given values into the formula:

$r_1+r_2=25+35=60\text{ cm}$

$\text{Length of TCT}=\sqrt{{80}^2-{60}^2}$

$\text{Length of TCT}=\sqrt{6400-3600}=\sqrt{2800}=20\sqrt{7}\text{ cm}$

Step 3: Calculate the length of the Direct Common Tangent (DCT)

The formula for the length of a direct common tangent is given by:

$\text{Length of DCT}=\sqrt{d^2-{(r_2-r_1)}^2}$

Substitute the given values into the formula:

$r_2-r_1=35-25=10\text{ cm}$

$\text{Length of DCT}=\sqrt{{80}^2-{10}^2}$

$\text{Length of DCT}=\sqrt{6400-100}=\sqrt{6300}=30\sqrt{7}\text{ cm}$

Step 4: Find the ratio of TCT to DCT

$\text{Ratio}=\frac{\text{Length of TCT}}{\text{Length of DCT}}=\frac{20\sqrt{7}}{30\sqrt{7}}$

Canceling out $\sqrt{7}$ and simplifying the fraction:

$\text{Ratio}=\frac{20}{30}=\frac{2}{3}$

Thus, the required ratio is $2:3$.