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  • If the curves  <img alt="\mathrm{a x^2+4 x y+2 y^2+x+y+5=0 \text { and } a x^2+6 x y+5 y^2+2 x+3 y+8=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Ba%20x%5E2&plus;4%2

If the curves  \mathrm{a x^2+4 x y+2 y^2+x+y+5=0 \text { and } a x^2+6 x y+5 y^2+2 x+3 y+8=0}  intersect at four

concyclic points then the value of a is

Option: 1

4


Option: 2

-4


Option: 3

6


Option: 4

-6


Answers (1)

best_answer

Any second degree curve passing through the intersection of the given curves is

\mathrm{a x^2+4 x y+2 y^2+x+y+5+\lambda\left(a x^2+6 x y+5 y^2+2 x+3 y+8\right)=0}

If it is a circle, then coefficient of \mathrm{ y^2} and coefficient of \mathrm{ xy=0} 

                        \mathrm{a(1+\lambda)=2+5 \lambda \text { and } 4+6 \lambda=0}

\mathrm{\Rightarrow \quad \mathrm{a}=\frac{2+5 \lambda}{1+\lambda} \text { and } \lambda=-\frac{2}{3} \Rightarrow \mathrm{a}=\frac{2-\frac{10}{3}}{1-\frac{2}{3}}=-4}

Hence (b) is correct answer.

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Gunjita

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