Question : The radii of two concentric circles are 17 cm and 25 cm. A straight line PQRS intersects the larger circle at the points P and S and intersects the smaller circle at the points Q and R. If QR = 16 cm, then the length (in cm) of PS is:
Option 1: 41
Option 2: 32
Option 3: 33
Option 4: 40
Correct Answer: 40
Solution :
Given:
OQ = radius of smaller circle = 17 cm
OP = radius of bigger circle = 25 cm
QR = 16 cm
OD is perpendicular to QR.
So, QD = RD = $\frac{16}{2}=8$ cm
From ∆ OQD,
OQ
2
= OD
2
+ QD
2
⇒ OD
2
= $17^2-8^2$
⇒ OD = $\sqrt{17^2-8^2}$
$\therefore$ OD = 15 cm
From ∆ OPD,
OP
2
= OD
2
+ PD
2
⇒ PD
2
= $25^2-15^2$
⇒ OD = $\sqrt{25^2-15^2}$
$\therefore$ PD = 20 cm
$\therefore$ PS = 2 × PD = 2 × 20 = 40 cm
Hence, the correct answer is 40.
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