Given that for the circle $x^{2} + y^{2} - 4 x + 6 y + 1 = 0$ , the line with equation $3 x - y$ is a chord. The middle point of the chord is

(\frac{2}{5}, \frac{11}{5})

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(\frac{- 2}{5}, \frac{11}{5})

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(\frac{- 2}{5}, \frac{- 11}{5})

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(\frac{2}{5} \frac{- 11}{5})

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Solution

The correct option is C $(\frac{- 2}{5}, \frac{- 11}{5})$
$3 x - y = 1$ is a chord of $x^{2} + y^{2} - 4 x + 6 y + 1 = 0$
Eqn of AB is $3 x - y = 1$
By property,
Perpendicular drawn from centre of circle bisects the chord
so, M is a foot of perpendicu;or from $O (2, - 3)$ on AB

$\frac{x - 2}{3} = \frac{y + 3}{- 1} = - \frac{(8)}{9 + 1} = \frac{- 4}{5}$
$\Rightarrow x = \frac{- 2}{5}$ and $y = \frac{- 11}{5}$

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Given that for the circle x2+y2−4x+6y+1=0 , the line with equation 3x−y is a chord. The middle point of the chord is

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Given that for the circle $x^{2} + y^{2} - 4 x + 6 y + 1 = 0$ , the line with equation $3 x - y$ is a chord. The middle point of the chord is