Question

At one end of a diameter PQ of a circle of radius 5 cm, tangent XPY is drawn to the circle. The length of chord AB parallel to XY and at a distance of 8 cm from P is

Answer 1

As we know that tangent at any point of a circle is perpendicular to the radius through the point of contact.
∴∠OAY=90°
As sum of cointerior angle is 180°.
Therefore,
∠OAY+∠OED=180°
⇒∠OED=90°
AE=8cm(From fig.)
Now in △OEC, by pythagoras theorem,
OC2=OE2 +EC2

⇒EC2 =OC2 −OE2

⇒EC2
=(5)2 −(3)2

⇒EC= 25−9
​
=4
Therefore,
Length of chord CD=2×CE(∵perpendicular from centre to the chord bisects the chord)
⇒CD=2×4=8cm
Hence the length of the chord CD is 8cm.