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# Find curve in $x-y$ plane which passes through$(1,1)$ . Intersects line $y=x$ at right angle at that point and satisfies $xy''+2y''=0$

$(a)\;y=\frac{1}{x^2} \\ (b)\;y = \frac{1}{x} \\ (c)\;y=0 \\ (d)\;y=x^2$

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A)
$y'= t$
$y''=t'$
$xt'+2t=0$
$x \large\frac{dt}{dx}$$=-2t \large\frac{dt}{t}=\frac{-2 dx}{x} \log t = -2 \log x +\log c t= \large\frac{c}{x^2} \large\frac{dy}{dx}=\frac{c}{x^2} dy =\large\frac{c dx}{x^2} y= \large\frac{-c}{x}$$+c_1$
It satisfies $(1,1)$
$1=-c+c_1$
Curve intersects the line $y=x$ at right angle means it is normal to curve.
So, $\large\frac{dy}{dx}=\frac{c}{x^2}$$=-1$
$c=-1$
So, $c_1=0$
So, $y= \large\frac{1}{x}$
Hence b is the correct answer.