A chord AB of a circle C1 of radius (√3 + 1) cm touches a circle C2 which is concentric to C1. If the radius of C2 is (√3 - 1) cm, the length of AB is: (SSC CGL 1st Sit. 2013)a)4√3 cmb)23 cmc)8√3 cmd)43 cmCorrect answer is option 'D'. Can you explain this answer?

Question

A chord AB of a circle C1 of radius (&radic;3 + 1) cm touches a circle C2 which is concentric to C1. If the radius of C2 is (&radic;3 - 1) cm, the length of AB is:  (SSC CGL 1st Sit. 2013)a)4&radic;3 cmb)23 cmc)8&radic;3 cmd)43 cmCorrect answer is option 'D'. Can you explain this answer?

Answer

Let A chord AB of circle C₁, touches the concentric circle C₂ at point ‘P’.

Here OA = radius of circle C₁ = (√3 + 1)
OP = radius of circle C₂ = (√3 - 1)
As AP is a tangent to the circle C₂.
∴ ∠OPA = 90°
Now, from ΔOPA, (OA)² = (OP)² + (AP)²
(AP)² = (OA)² – (OP)²

Answer

Understanding the Problem
We have two concentric circles, C1 and C2. The radius of circle C1 is $ \sqrt{3} + 1 $ cm, and the radius of circle C2 is $ \sqrt{3} - 1 $ cm. A chord AB of circle C1 touches circle C2.
Key Pointers
- Radius of C1: $ r_1 = \sqrt{3} + 1 $ cm
- Radius of C2: $ r_2 = \sqrt{3} - 1 $ cm
- Distance from Center to Chord: The distance from the center of the circles to chord AB is equal to the radius of circle C2, which is $ r_2 $.
Applying the Chord Length Formula
The length of a chord $ AB $ in a circle can be calculated using the formula:
$$
\text{Length of AB} = 2 \sqrt{r_1^2 - d^2}
$$
where:
- $ r_1 $ is the radius of the circle (C1)
- $ d $ is the distance from the center to the chord (equal to the radius of C2, $ r_2 $)
Calculating $ r_1^2 $ and $ d^2 $
- Calculate $ r_1^2 $:
$$
r_1^2 = (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}
$$
- Calculate $ d^2 $:
$$
d^2 = (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}
$$
Substituting Values in the Formula
Now substitute $ r_1^2 $ and $ d^2 $ into the chord length formula:
$$
\text{Length of AB} = 2 \sqrt{(4 + 2\sqrt{3}) - (4 - 2\sqrt{3})}
$$
This simplifies to:
$$
\text{Length of AB} = 2 \sqrt{4\sqrt{3}} = 2 \cdot 2\sqrt{3} = 4\sqrt{3} \text{ cm}
$$
Conclusion
Thus, the length of chord AB is $ \textbf{4\sqrt{3} cm} $. The correct answer is option 'a'.

Similar SSC CGL Doubts

A chord of length 30 cm is at a distance of 8 cm from the centre of a circle. The radius of the circle is: (SSC CGL 1st Sit. 2013)

N is the foot of the perpendicular from a point P of a circle with radius 7 cm, on a diameter AB of the circle. If the length of the chord PB is 12 cm, the distance of the point N from the point B is (SSC CGL 1st Sit. 2013)

Two circles touch each other internally. Their radii are 2 cm and 3 cm. The biggest chord of the greater circle which is outside the inner circle of length. (SSC CGL 2nd Sit. 2013)

AB and CD are two parallel chords of a circle such that AB = 10 cm and CD = 24 cm. If the chords are on the opposite sides of the centre and distance between them is 17 cm, then the radius of the circle is: (SSC CGL 1st Sit. 2013)

In a circle of radius 17 cm, two parallel chords of lengths 30 cm and 16 cm are drawn. If both the chords are on the same side of the centre, then the distance between the chords is (SSC CGL 2nd Sit. 2013)

Top Courses for SSC CGLView all

General Awareness for SSC CGL

Lucent for GK

Quantitative Aptitude for SSC CGL

Current Affairs & Hindu Analysis: Daily, Weekly & Monthly

How to Prepare for SSC CGL

Top Courses for SSC CGL

Top Courses for SSC CGL

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams