SSC CHSL 2024 PYQ — Question 70

The correct answer is option 1 i.e. 2 : 3.

The radius of the first circle (r₁) = 25 cm

The radius of the second circle (r₂) = 35 cm

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

Length of the direct common tangent (L_D) = $\sqrt{d^2 - (r_1 - r_2)^2}$

 = $\sqrt{80^2 - (35 - 25)^2} = \sqrt{6400 - 100} = \sqrt{6300}$ = 30$\sqrt7$

Length of the transverse common tangent (L_T) = $\sqrt{d^2 - (r_1 + r_2)^2}$

⇒ L_T =  $= \sqrt{80^2 - (35 + 25)^2}$

⇒ $\sqrt{6400 - 3600} = \sqrt{2800}$ = 20$\sqrt7$

Ratio =  L_T /L_D = 20$\sqrt7$/ 30$\sqrt7$ = 2/3

The ratio of the lengths of the transverse common tangent to the direct common tangent is approximately 2:3.