Question

Let a curve $y=f\left(x\right)$, $f\left(x\right)\ge 0,\forall xϵR$ has property that for every point $P$ on the curve length of subnormal is equal to abscissa of $P$. If $f\left(1\right)=3$, then $f\left(4\right)$ is equal to

• A. $-2\sqrt{6}$
• B. $2\sqrt{6}$
• C. $3\sqrt{5}$
• D. none of these

Answer 1

Answer: (B)

$2\sqrt{6}$

It is given that
$|y.f^{\prime }\left(x\right)|=x$
And $f\left(x\right)\ge 0$.
Hence
$f^{\prime }\left(x\right)=\frac{x}{y}$.
Or
$\frac{dy}{dx}=\frac{x}{y}$.
Or
$y.dy=x.dx$
Integrating,
$\frac{y^2}{2}=\frac{x^2}{2}+c$
Now
$f\left(1\right)=3$.
Hence
$9=1+2c$
Or
$c=4$
Hence
$y^2=x^2+8$
Hence
$f(4{)}^2=16+8$
Or
$f(4{)}^2=24$
Or
$f\left(4\right)=2\sqrt{6}$ since $f\left(x\right)\ge 0$ for all x.