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Question

Let tangents are drawn at two points of the circle (x7)2+(y+1)2=25. If the point of intersection of both the tangents is origin, then the angle between them (in degrees) is

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Solution

Any line passing through origin is written as
ymx=0
This line is tangent to the circle (x7)2+(y+1)2=25, so the distance from centre to the line is equal to radius,
|17m|1+m2=5
Squaring on both the sides, we get
(1+7m)2=25(1+m2)
24m2+14m24=0D=142+4×242>0
Let the roots of the equation be m1,m2, so
m1m2=2424=1

Hence, the angle between the tangents is 90


Alternate solution :
Given circle is (x7)2+(y+1)2=25


Distance between OC
=72+12=50=52
From the figure, we get
sinθ2=CPOCsinθ2=552=12θ2=45θ=90

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