Question

The curve, with the property that the projection of the ordinate on the normal is constant and has a length equal to ' $a$ ', is

• A. $a{\log }_e\left|y+\sqrt{y^2-a^2}\right|=x+c$
• B. $a{\log }_e\left|x+\sqrt{x^2-a^2}\right|=y+c$
• C. $ay={\tan }^{-1}(x+c)$
• D. None of these

Answer 1

Answer: (A)

Let $P\equiv (x,y)$ on the curve $y=f(x)$
Ordinate is $PM,$ where $M$ is the foot of the perpendicular from point $P$ on the $x$ -axis
Projection of ordinate on normal $=P{M}^{\prime }$
$\therefore P{M}^{\prime }=PM\cos \theta =a$ (given)

$\therefore \frac{y}{\sqrt{1+{\tan }^2\theta }}=a$
$\Rightarrow y=a\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$
$\Rightarrow \frac{dy}{dx}=\frac{\sqrt{y^2-a^2}}{a}$
$\Rightarrow \int \frac{ady}{\sqrt{y^2-a^2}}=\int dx$
$\Rightarrow a{\log }_e|y+\sqrt{y^2-a^2}\mid =x+c$