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  4. If P is a point on the parabola y=x2+4 which is closest to the straight line y =4 x -1, then the co-ordinates of P are

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Q. If $P$ is a point on the parabola $y=x^{2}+4$ which is closest to the straight line $y =4 x -1$, then the co-ordinates of $P$ are

JEE MainJEE Main 2021Conic Sections

Solution:

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$P: y=x^{2}+4$
$k=h^{2}+4$
$L : y=4 x-1$
$y-4 x+1=0$
$d=A B=\left|\frac{k-4 h+1}{\sqrt{5}}\right|$
$=\left|\frac{h^{2}-4-4 h+1}{\sqrt{5}}\right|$
$\frac{d(d)}{d h}=\frac{2 h-4}{\sqrt{5}}=0$
$h=2$
$\frac{d^{2}(d)}{d h^{2}}=\frac{2}{\sqrt{5}}>0$
$\therefore k=4+4=8$
$\therefore $ Point (2,8)