The tangent to the circle C1:x2+y2−2x−1=0 at the point (2,1) cuts off a chord of length 4 from a circle C2 whose centre is (3,−2). The radius of C2 is
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Official solution
Here, equation of tangent on C1 at (2, 1) is :
2x + y - (x + 2) -1 = 0
Or x + y = 3
If it cuts off the chord of the circle C2 then the equation of the chord is :
x + y = 3
\[\therefore\,\,\,\] distance of the chord from (3, - 2) is :
d = \[\left| {{{3 - 2 - 3} \over {\sqrt 2 }}} \right|\] = \[\sqrt 2 \]
Also, length of the chord is l = 4
\[\therefore\,\,\,\] radius of C2 = r = \[\sqrt {{{\left( {{l \over 2}} \right)}^2} + {d^2}} \]
= \[\sqrt {{{\left( 2 \right)}^2} + {{\left( {\sqrt 2 } \right)}^2}} = \sqrt 6 \]