CAT 2020 Slot 2 QA Question 22: Let C1 and C2 be concentric circles such that the diameter of C1 is 2 cm longer than that of C2. If a chord of C1 has length 6 cm and is a tangent to C2, then the diameter, in cm, of C1 is

Question

Let C1 and C2 be concentric circles such that the diameter of C1 is 2 cm longer than that of C2. If a chord of C1 has length 6 cm and is a tangent to C2, then the diameter, in cm, of C1 is

Accepted Answer

Let the radii of C1 and C2 be $R$ and $r$ respectively.

The diameter of C1 is $2$ cm more than the diameter of C2, so
$2R - 2r = 2$
$R - r = 1$
$R = r + 1$

The chord of C1 is tangent to C2, so its distance from the common center is $r$. Half the chord is $3$ cm.
$R^2 = r^2 + 3^2$
$(r+1)^2 = r^2 +