# If the lines $\dpi{100} 2x+3y+1=0\; and\; 3x-y-4=0$  lie along diameters of a circle of circumference $\dpi{100} 10\pi ,$ then the equation of the circle is Option 1) $x^{2}+y^{2}+2x+2y-23=0$ Option 2) $x^{2}+y^{2}-2x-2y-23=0$ Option 3) $x^{2}+y^{2}-2x+2y-23=0$ Option 4) $x^{2}+y^{2}+2x-2y-23=0$

P Plabita

As we learnt in

General form of a circle -

$x^{2}+y^{2}+2gx+2fy+c= 0$

- wherein

centre = $\left ( -g,-f \right )$

radius = $\sqrt{g^{2}+f^{2}-c}$

coordinates of centre is the point of  intersection of 2x+3y + 1 =0 and 3x - y - 4 = 0

$2x+3y=-1$

$\underline{9x-3y=12}$

$11x=11\Rightarrow x=1$

$2(1)+3y=-1$

$y=-1$

$Point\ is (1,-1)$

$also, 2\pi r=10\pi$

$\Rightarrow r=5$

Equation is

$(x-1)^{2}+(y+1)^{2}=5^{2}\\ x^{2}+y^{2}-2x+2y-23=0$

Option 1)

$x^{2}+y^{2}+2x+2y-23=0$

This option is incorrect

Option 2)

$x^{2}+y^{2}-2x-2y-23=0$

This option is incorrect

Option 3)

$x^{2}+y^{2}-2x+2y-23=0$

This option is correct

Option 4)

$x^{2}+y^{2}+2x-2y-23=0$

This option is incorrect

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