Question : PQ is a chord of length 8 cm of a circle with centre O and radius 5 cm. The tangents at P and Q intersect at a point T. The length of TP is:
Option 1: $\frac{20}{3}\ \text{cm}$
Option 2: $\frac{21}{4}\ \text{cm}$
Option 3: $\frac{10}{3}\ \text{cm}$
Option 4: $\frac{15}{4}\ \text{cm}$
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Correct Answer: $\frac{20}{3}\ \text{cm}$
Solution : Join OP and OT. Let OT intersect PQ at a point R. Then, TP = TQ (The lengths of tangents drawn from an external point to a circle are equal) and $\angle$PTR = $\angle$QTR. ⇒ TR $\perp$ PQ and TR bisect PQ. ⇒ PR = RQ = 4 cm OP = 5 cm So, OR2 = OP2 – PR2 = $\sqrt{5^2-4^2}=\sqrt{9}=3$ cm Let TP = $x$ cm and TR = $y$ cm From right ΔTRP, we get, TP2 = TR2 + PR2 ⇒ $x$2 = $y$2 + 16 ⇒ $x$2 − $y$2 = 16-------------------------(i) From right ΔOPT, we get TP2 + OP2 = OT2 ⇒ $x$2 + 52 = ($y$ + 3)2 [$\because$ OT2 = (OR + RT)2 ] ⇒ $x$2 − $y$2 = 6$y$ − 16------------------(ii) From (i) and (ii), we get, 6$y$ − 16 = 16 ⇒ 6$y$ = 32 ⇒ $y$ = $\frac{16}{3}$ Putting the value of $y$ in equation (i), we get, $x$2 $=16 + (\frac{16}{3})^2=16+\frac{256}{9}=\frac{400}{9}=\frac{20}{3}\ \text{cm}$ Hence, the correct answer is $\frac{20}{3}\ \text{cm}$.
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Question : The radius of a circle is 10 cm. The distance of chord PQ from the centre is 8 cm. What is the length of chord PQ?
Option 1: 16 cm
Option 2: 15 cm
Option 3: 12 cm
Option 4: 14 cm
Question : In a circle with a centre at O(0,0) and a radius of 5 cm, AB is a chord of length 8 cm. If OM is perpendicular to AB, then the length of OM is:
Option 1: 2.5 cm
Option 2: 3 cm
Option 3: 4 cm
Option 4: 1 cm
Question : The diameter $PQ$ of a circle with centre $O$ is perpendicular to the chord $RS$. $PQ$ intersects $RS$ at $T$. If $RS=16$ cm and $QT=4$ cm, what is the length (in cm) of the diameter of the circle?
Option 1: 20
Option 2: 48
Option 3: 24
Option 4: 10
Question : The angle subtended by a chord PQ on the centre of a circle is 180°. If the length of chord PQ is 54 cm, then what will be the diameter of this circle?
Option 1: 54 cm
Option 2: 81 cm
Option 3: 27 cm
Option 4: 64 cm
Question : A chord of length 24 cm is at a distance of 5 cm from the centre of the circle. The radius of the circle is:
Option 1: 13 cm
Option 2: 12 cm
Option 3: 16 cm
Option 4: 15 cm
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