(A) Let $r$ be the radius of the circle. The center of the circle is $C(2,4)$.The perpendicular distance $d$ from the center $C(2,4)$ to the line $x+y

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(A) Let $r$ be the radius of the circle. The center of the circle is $C(2,4)$.The perpendicular distance $d$ from the center $C(2,4)$ to the line $x+y+2=0$ is given by:$d = \frac{|2+4+2|}{\sqrt{1^2+1^2}} = \frac{8}{\sqrt{2}} = 4\sqrt{2}$.The chord length is $6$,so the half-chord length is $AB = \frac{6}{2} = 3$.In the right-angled triangle $\triangle CAB$,by the Pythagorean theorem:$r^2 = d^2 + (AB)^2$$r^2 = (4\sqrt{2})^2 + 3^2$$r^2 = 32 + 9 = 41$$r = \sqrt{41}$.

Class 11 Mathematics 10-1.Circle and System of Circles Geometrical problems regarding circle and its properties

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$A$ circle with center at $(2,4)$ is such that the line $x+y+2=0$ cuts a chord of length $6$. The radius of the circle is

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A
$\sqrt{41}$
B
$\sqrt{11}$
C
$\sqrt{21}$
D
$\sqrt{31}$

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10-1.Circle and System of Circles

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Geometrical problems regarding circle and its properties

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