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  4. If radius of circle passing through the focus of parabola x2=4 y and touches it at M(6,9) is p √10(p ∈ N), then find p.

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Q. If radius of circle passing through the focus of parabola $x^2=4 y$ and touches it at $M(6,9)$ is $p \sqrt{10}(p \in N)$, then find $p$.

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Solution:

$(x-6)^2+(y-9)^2+\lambda(3 x-y-9)=0$
image
Satisfy $(0,1)$ gives $\lambda=10$
$\Rightarrow x^2+y^2+18 x-28 y+27=0, r=5 \sqrt{10}$
$\Rightarrow p=5 $