In Fig. 8.4, the pair of tangents AP and AQ drawn from an external point A to a circle with centre O are perpendicular to each other and length of each tangent is 5 cm. Then radius of the circle is
A circle touches x-axis at A and y-axis at B. If O is origin and OA = 5 units, then diameter of the circle is
OA = OB ⇒ OB = 5
AC = BC [Radii]
⇒ OACB is a square.
⇒ AC = OA = 5
⇒ Diameter =10 units.
At point A on a diameter AB of a circle of radius 10 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY at a distance 16 cm from A is
In the given figure
XY is the tangent to the circle
We know that tangent is at right angle with the line joining the tangent point to the centre of the circle
Therefore,
∠OAY = 90°
∵ CD is parallel to XY
Therefore,
∠OAY = ∠BED = 90° (Corresponding angles)
Again,
Given that
AE = 16 cm
And the radius of the circle is 10 cm
Therefore,
AO = 10 cm
∴ OE = AE - AO = 16 - 10 = 6 cm
Also,
OC = 10 cm (Radius of the circle)
From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is
Two circles touch each other externally at C and AB is common tangent of circles, then ∠ACB is
In figure AT is a tangent to the circle with centre O such that OT = 4 cm and ∠OTA = 30°. Then AT is equal to
∠OAT = 90° [∵ Tangent and radius are ⊥ to each other at the point of contact
In right-angle ΔOAT
In Fig. 8.8, if PA anti PB are tangents to the circle with centre O such that ∠APB = 50°, then ∠OAB is equal to
In figure if O is centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to
OP ⊥ PR [∵ Tangent and radius are ⊥ to each other at the point of contact]
∠OPQ = 90° - 50° = 40°
OP - OQ [Radii]
∴ ∠OPQ = ∠OQP = 40°
In ΔOPQ,
⇒ ∠POQ + ∠OPQ + ∠OQP = 180°
⇒ ∠POQ + 40° + 40° = 180°
∠POQ = 180° - 80° = 100°.
If ΔABC is circumscribing a circle in the Fig. 8.12. The length of AB is
From a point X, the length of the tangent to a circle is 20 cm and the distance of X from the centre is 25 cm. The radius of the circle is
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