Question

Two circles touch each other internally at a single point. Their radii are 2 cm and 3 cm. The biggest chord of outer circle which is a tangent to the inner circle, is of length ____.

• A. 2$\sqrt{2}$ cm
• B. 3$\sqrt{2}$ cm
• C. 2$\sqrt{3}$ cm
• D. 4$\sqrt{2}$cm

Answer 1

The correct option is D

4$\sqrt{2}$cm

Given that OD = 2 cm

⇒ CD = 2(OD) = 4 cm

PD = 3 cm

Therefore, CP = CD - PD = 1 cm
Now, as tangent is perpendicular to radius at point of contact, applying pythagoras theorem for triangle CPA, we have:

${CA}^2$ = ${PA}^2$ - ${PC}^2$
${CA}^2$ = $3^2$ - $1^2$
CA = $\sqrt{8}$ cm

As CP is perpendicular to chord AB, hence it bisects the chord AB.
i.e., BA = 2 CA = 2$\sqrt{8}$ = 4$\sqrt{2}$ cm

Therefore, length of the biggest chord is 4$\sqrt{2}$ cm.