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 option is A

Given:

Radius of the first circle (r₁) = 25 cm

Radius of the second circle (r₂) = 35 cm

Distance between the centers of the circles (d) = 80 cm

Formula used:

The length of the direct common tangent (D) between two circles is given by the formula:

D = √(d² - (r₁ - r₂)²)

The length of the transverse common tangent (T) between two circles is given by the formula:

T = √(d² - (r₁ + r₂)²)

​Solution:​

 Calculate the length of the direct common tangent

=> D = √(80² - (35 - 25)²)

=> D = √(6400 - 10²)

=> D = √(6400 - 100)

=> D = √6300 cm

Calculate the length of the transverse common tangent

=> T = √(80² - (35 + 25)²)

=> T = √(6400 - 60²)

=> T = √(6400 - 3600)

=> T = √2800 cm

=> Ratio = T / D = √2800 : √6300 = √(2800/6300) = √(28/63) = √4/9 = 2/3 = 2 : 3