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. $a y=\tan ^{-1}(x+c)$
• D. None of these

Answer 1

Answer: (A)

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

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