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. $0$

Answer 1

Answer: (B)

$2\sqrt{6}$

Given $y\frac{dy}{dx}=x$
$\Rightarrow ydy=xdx$
$\Rightarrow y^2=x^2+c$
Also, given $f\left(1\right)=3$
$\Rightarrow c=8$
$\Rightarrow y^2=x^2+8,$
or, $f\left(x\right)=y=\sqrt{x^2+8}$
$\Rightarrow f\left(4\right)=\sqrt{16+8}=2\sqrt{6}$