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  • If the curves <img alt="\mathrm{a x^2+4 x y+2 y^2+x+y+5=0 \: \: 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%20x%2

If the curves \mathrm{a x^2+4 x y+2 y^2+x+y+5=0 \: \: 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{ x^2= } coefficient of \mathrm{ y^2 } and coefficient of \mathrm{ x y=0 }

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

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

\mathrm{ \Rightarrow a=\frac{2-\frac{10}{3}}{1-\frac{2}{3}}=-4 }
Hence option 2 is correct.


 

Posted by

Rishi

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