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  • Radius of the circle that passes through origin and touches the parabola at t

Radius of the circle that passes through origin and touches the parabola \mathrm{y^2=4 a x} at the point \mathrm{(a, 2 a)} is
 

Option: 1

\mathrm{\frac{5}{\sqrt{2}} a}
 


Option: 2

\mathrm{2 \sqrt{2} a}
 


Option: 3

\mathrm{5 \sqrt{2} a}
 


Option: 4

\mathrm{3 \sqrt{2} a}


Answers (1)

best_answer

Equation of tangent of parabola at \mathrm{ (a, 2 a)} is

\mathrm{ y \cdot 2 a=2 a(x+a) \Rightarrow y-x-a=0 }

Equation of circle touching the parabola at \mathrm{ (a, 2 a)} is

\mathrm{ (x-a)^2+(y-2 a)^2+\lambda(y-x-a)=0 }

It passes through \mathrm{ (0,0) }

\mathrm{ \therefore a^2+4 a^2+\lambda(-a)=0 \Rightarrow \lambda=5 a \text {. } }

\mathrm{ Thus, required\: circle\: is \: x^2+y^2-7 a x+a y=0, \: its \: radius \: is \sqrt{\frac{49}{4} a^2+\frac{a^2}{4}}=\frac{5}{\sqrt{2}} a }
Hence option 1 is correct.

Posted by

manish

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