If a chord of the circle x 2 + y 2 − 4 x − 2 y − c = 0 is trisected at the points ( 3 1 , 3 1 ) and ( 3 8 , 3 8 ) , then c =

Let the points be A and B and the centre C ( 2 , 1 ) . The chord lies on y = x and A B = 3 7 2 , so the whole chord is PQ = 7 2 . The distance of C from y = x is 2 1 . Then r 2 = ( 2 7 2 ) 2 + ( 2 1 ) 2 = 25 , and r 2 = 4 + 1 + c , so c = 20 .

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MathematicsCircleChord, Intercept (Made by the circle)Medium2 minQB

MathematicsMediumsingle choice

If a chord of the circle $x^{2} + y^{2} - 4 x - 2 y - c = 0$ is trisected at the points $(\frac{1}{3}, \frac{1}{3})$ and $(\frac{8}{3}, \frac{8}{3})$ , then $c =$

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Let the points be $A$ and $B$ and the centre $C (2, 1)$ . The chord lies on $y = x$ and $A B = \frac{7 2}{3}$ , so the whole chord is $PQ = 72$ . The distance of C from $y = x$ is $\frac{1}{2}$ . Then $r^{2} = (\frac{7 2}{2})^{2} + (\frac{1}{2})^{2} = 25$ , and $r^{2} = 4 + 1 + c$ , so $c = 20$ .

Stream:JEESubject:MathematicsTopic:CircleSubtopic:Chord, Intercept (Made by the circle)

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