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Question

AP = x, PC = y and OP = r, the in radius of the right angled triangle ABC, with B right angled.

Find the value of (1 + rx)(1 + ry)

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Solution

Observe that quadrilateral OQBR has QBR = 90.

Since AB and BC are tangents to the circle, OQ and OR are perpendicular to AB and BC respectively and so BQO and BRO are also 90. Since three angles in a quadrilateral are 90, the fourth angle will be 360 - 3( 90) = 90. So it is a rectangle. Also, since OQ = OR, it follows that all the sides are equal and hence quadrilateral OQBR is actually a square. So, BQ = BR = r.

(x+y)2 = (r+x)2 + (r+y)2

x2+y2+2xy=r2+x2+2rx+r2+y2+2ry

2xy=2r2+2rx+2ry

xy=r2+r(x+y)

r2+r(x+y)+xy=xy+xy

(r+x)(r+y)=2xy

(r+x)(r+y)xy = 2

( r+xx)( r+yy) = 2

(1 + rx)(1 + ry) = 2


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