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Q6 of 47 Page 10

Find the locus of centres of circles which touch two intersecting lines.

In Δ YXO and Δ ZXO,
∠ OYX = ∠ OZX (= 90°)
XO = XO (Common)
OY = OZ = radius
Δ YOX ≅ Δ ZOX (By SAS congruence)
∴ ∠ YXO = ∠ ZXO
∴ Locus of the centre is a straight line bisecting them between two intersecting lines.

More from this chapter

4 In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length. 5 Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord. 7 Let A be one point of intersection of two intersecting circles with centres O and Q. The tangents at A to the two circles meet the circles again at B and C, respectively. Let the point P be located so that AOPQ is a parallelogram. Prove that P is the circumcentre of the triangle ABC in figure.
(Hint: AQ ⊥ AB and AQ || OP. Then OP ⊥ AB and is also bisector of AB. Similarly, PQ is perpendicular bisector of AC.)

8 The radius of the incircle of a triangle is 4 cm and the segments into which one side is divided by the point of contact are 6 cm and
8 cm. Determine the other two sides of the triangle.
(Hint: Equate the areas of the triangle found by using the formula √[s(s-a)(s-b)(s-c)] and also found by dividing it into three triangles.)