Q3 of 31 Page 233

is a diameter of ⨀(O, 10). A tangent is drawn from B to ⨀(O, 8) which touches ⨀(O, 8) at D. intersects 0(0, 10) in C. Find AC.


Given that AB is a diameter of circle (O, 10).


⇒ OA = OB = 10 = radius


⇒ AB = 20 = diameter


Also given a tangent is drawn from B to circle (O, 8) which touches the circle at D.


⇒ OD = 8 = radius


Since BD is a tangent, OD ⊥ BD.


And since the angle is inscribed in a semi circle, ∠ACB = 90°.


⇒ ∠ODB = ∠ACB = 90°


∴ ∠DBO ≅ ∠CBA


By AA theorem,


Thus, correspondence ODB ↔ ACB is a similarity.


Then =


⇒ =


⇒ AC = = 16


∴ AC = 16


More from this chapter

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1

P is the point in the exterior of ⨀(O, r) and the tangents from P to the circle touch the circle at X and Y.

(1) Find OP, if r = 12, XP = 5


(2) Find m∠XPO, if m∠XOY = 110


(3) Find r, if OP = 25 and PY = 24


(4) Find m∠XOP, if m∠XPO = 80

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Two concentric circles having radii 73 and 55 are given. The chord of the circle with larger radius touches the circle with smaller radius. Find the length of the chord.

 4

P is in the exterior of a circle at distance 34 from the centre 0. A line through P touches the circle at Q. PQ = 16, find the diameter of the circle.

 5

In figure 11.24, two tangents are drawn to a circle from a point A which is in the exterior of the circle. The points of contact of the tangents are P and Q as shown in the figure. A line 1 touches the circle at R and intersects and in B and C respectively. If AB = c, BC = a, CA = b, then prove that

(1) AP + AQ = a + b + c


(2) AB + BR = AC + CR = AP = AQ =