Question

If at any point on a curve the subtangent and subnormal are equal, then the tangent is equal to:

• A. ordinate
• B. $\sqrt{2}$ ordinate
• C. $\sqrt{2(\text{ordinate})}$
• D. none of these

Answer 1

Answer: (B)

we have, subtangent = subnormal
$\Rightarrow \frac{y}{\frac{dy}{dx}}=y.\frac{dy}{dx}$
$\Rightarrow {\left(\frac{dy}{dx}\right)}^2=1\Rightarrow \frac{dy}{dx}=\pm 1$ .....(1)
So, length of the tangent $=\frac{y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}}{\frac{dy}{dx}}$
$\Rightarrow$ length of the tangent = $\sqrt{2}y$
[using equation (1)]
$\Rightarrow$ length of the tangent = $\sqrt{2}$ ordinate