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  4. 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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Q. The tangent to the circle $C_1 : x^2 + y^2 - 2x - 1 = 0$ at the point $(2, 1)$ cuts off a chord of length $4$ from a circle $C_2$ whose centre is $(3, -2)$. The radius of $C_2$ is :

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Solution:

Equation of tangent to circle $x^{2}+y^{2}-2 x-1=0$ at point $(2,1)$ is
$2 x+y-(x+2)-1=0 $
$x+y-3=0$
This line is chord of circle $C_{2}$
image
So, $O C=\frac{|3-2-3|}{\sqrt{2}}=\sqrt{2}$
Therefore, radius of circle is $\sqrt{C A^{2}+O C^{2}}=\sqrt{4+2}=\sqrt{6}$