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  • If a circle passes through the points of intersection of the coordinate axes with the lines <img alt="\mathrm{\lambda x y+1=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7

If a circle passes through the points of intersection of the coordinate axes with the lines \mathrm{\lambda x y+1=0} and \mathrm{x-2 y+3=0} , then value of \mathrm{\lambda} is equal to

Option: 1

\mathrm{\frac{1}{2}}


Option: 2

2


Option: 3

3


Option: 4

6


Answers (1)

best_answer

Let the equation of the circle be

\mathrm{(\lambda x-y+1)(x-2 y+3)+k x y=0} -----------(1)

If it represents a circle, the conditions are 

        \mathrm{\left[\begin{array}{c} \text { coeff. of } x^2=\text { coeff. of } y^2 \\ \text { coeff.of } x y \text { is zero } \end{array}\right.}

Or 

        \mathrm{\left[\begin{array}{c} \lambda=2 \\ -2 \lambda-1+k=0 \end{array}\right.}

        \mathrm{\Rightarrow \lambda=2, k=5}

The equation of circle (1) becomes

\mathrm{2 x^2+2 y^2+7 x-5 y+3=0}

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manish

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