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  • Equation of the circle touching the circle <img alt="\mathrm{x^2+y^2-15 x+5 y=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%5E2&plus;y%5E2-15%20x&plus;5%20y%3D0%7D"

Equation of the circle touching the circle \mathrm{x^2+y^2-15 x+5 y=0} at the point \mathrm{(1,2)} and passing through the point \mathrm{(0,2)} is
 

Option: 1

\mathrm{13 x^2+13 y^2-13 x-61 y+70=0}


Option: 2

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


Option: 3

\mathrm{13 x^2+13 y^2-13 x-61 y+9=0}


Option: 4

none of these


Answers (1)

best_answer

 Let \mathrm{S_1=x^2+y^2-15 x+5 y=0} \quad \quad \quad \dots(i)

Any point circle through (1,2) is given as

\mathrm{S_2=(x-1)^2+(y-2)^2=0}\quad \quad \quad \dots(ii)
Family of circles passing through \mathrm{S_1=0} and \mathrm{S_2=0} is \mathrm{S_1+\lambda S_2=0}, where, \lambda is a parameter (\lambda \neq-1) \quad \quad \dots(iii)

Now, circle (iii) passes through \mathrm{(0,2) \Rightarrow \lambda=-14}

Putting, \mathrm{\lambda=-14} in (iii), we get the required equation.

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Rishi

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