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. $x+a\ln \left(\sqrt{y^2-a^2}+y\right)=c$
• B. $x+\sqrt{a^2-y^2}=c$
• C. $(y-a{)}^2=cx$
• D. $ay={\tan }^{-1}(x+c)$

Answer 1

Answer: (A)

$\text{ Ordinate }=PM.\text{ Let }P\equiv (x,y)[T/S]$
$\text{ Projection of ordinate on normal }=PN$
$PN=PM\cos \theta =a\text{ (given) }$
$\therefore \frac{y}{\sqrt{1+{\tan }^2\theta }}=a\Rightarrow y=a\sqrt{1+{\left(y_1\right)}^2}$
$\Rightarrow \frac{dy}{dx}=\frac{\sqrt{y^2-a^2}}{a}\Rightarrow \int \frac{ady}{\sqrt{y^2-a^2}}=\int dx\Rightarrow a\ln \left|y+\sqrt{y^2-a^2}\right|=x+c$