Question : Chord PQ is the perpendicular bisector of radius OA of the circle with centre O. (A is a point on the edge of the circle). If the length of Arc $PAQ=\frac{2\pi}{3}$. What is the length of chord PQ?
Option 1: $2$
Option 2: $\sqrt{3}$
Option 3: $2\sqrt{3}$
Option 4: $1$
Correct Answer: $\sqrt{3}$
Solution : PQ is perpendicular bisector of OA. $\therefore$ OP = OQ = PA = AQ $\therefore$ OPAQ is a rhombus. As we know the angle subtended at the centre by an arc is twice that at the circumference. Thus, 2 $\angle$PAQ = Reflex $\angle$POQ ⇒ 2 $\angle$PAQ = 360° – $\angle$POQ ⇒ 3$\angle$ PAQ = 360° ($\because\angle$PAQ = $\angle$POQ) $\therefore \angle$ PAQ = 120$^\circ$ = $\angle$ POQ = $\frac{2\pi}{3}$ Again, Radius (r) = $\frac{\text{arc length}}{\theta}$ = $\frac{\frac{2\pi}{3}}{\frac{2\pi}{3}}$ = 1 Now, in $\triangle$ OPB, OP = 1 unit $\angle$POB = 60° $\therefore$ sin 60° = $\frac{PB}{OP}$ ⇒ PB = $\frac{\sqrt3}{2}$ $\therefore$ PQ = 2 × $\frac{\sqrt3}{2}=\sqrt3$ Hence, the correct answer is $\sqrt3$.
Application | Eligibility | Selection Process | Result | Cutoff | Admit Card | Preparation Tips
Question : Out of two concentric circles, the radius of the outer circle is 6 cm and the chord PQ of the length 10 cm is a tangent to the inner circle. Find the radius (in cm) of the inner circle.
Option 1: $4$
Option 2: $\sqrt{7}$
Option 3: $\sqrt{13}$
Option 4: $\sqrt{11}$
Question : Let C be a circle with centre O and radius 5 cm. Let PQ be a tangent to the circle and A be the point of tangency. Let B be a point on PQ such that the length of AB is 12 cm. If the line joining O and B intersects the circle at R, find the length of BR (in cm).
Option 1: 2
Option 2: 13
Option 3: 6
Option 4: 8
Question : AB is a chord in a circle of radius 13 cm. From centre O, a perpendicular is drawn through AB intersecting AB at point C. The length of OC is 5 cm. What is the length of AB?
Option 1: 24 cm
Option 2: 12 cm
Option 3: 20 cm
Option 4: 15 cm
Question : The area of a circle is the same as the area of a square. What is the ratio of the diameter of the circle and the diagonal of the square?
Option 1: $1:\sqrt{\pi }$
Option 2: $2:\sqrt{\pi }$
Option 3: $\sqrt{2}:\sqrt{\pi }$
Option 4: $1:{\pi }$
Question : In a given circle, the chord PQ is of length 18 cm. AB is the perpendicular bisector of PQ at M. If MB = 3 cm, then the length of AB is:
Option 1: 27 cm
Option 2: 30 cm
Option 3: 28 cm
Option 4: 25 cm
Regular exam updates, QnA, Predictors, College Applications & E-books now on your Mobile