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  • The equation of the circle which touches the circle <img alt="\mathrm{x^2+y^2-6 x+6 y+17=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%5E2&plus;y%5E2-6%20x&plus;6%2

The equation of the circle which touches the circle \mathrm{x^2+y^2-6 x+6 y+17=0} externally and to which the lines

\mathrm{x^2-3 x y-3 x+9 y=0} are normals, is

Option: 1

\mathrm{x^2+y^2-6 x-2 y-1=0 }


Option: 2

\mathrm{x^2+y^2+6 x+2 y+1=0}


Option: 3

\mathrm{x^2+y^2-6 x-6 y+1=0}


Option: 4

\mathrm{x^2+y^2-6 x-2 y+1=0}


Answers (1)

best_answer

Joint equations of normals are

\mathrm{x^2-3 x y-3 x+9 y=0 \Rightarrow x(x-3 y)-3(x-3 y)=0 \Rightarrow(x-3)(x-3 y)=0}

\therefore \quad  Given normals are \mathrm{x-3=0} and \mathrm{x-3 y=0}, which intersect at centre of circle whose coordinates are \mathrm{(3,1)}.

The given circle is \mathrm{C_1=(3,-3), r_1=1 ; C_2=(3,1), r_2=}  ?

If the two circles touch externally, then \mathrm{C_1 C_2=r_1+r_2 \Rightarrow 4=1+r_2 \Rightarrow r_2=3}

\therefore Equation of required circle is \mathrm{(x-3)^2+(y-1)^2=(3)^2 \Rightarrow x^2+y^2-6 x-2 y+1=0}

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