Question

If at any point $(x_1,y_1)$ on a curve the subtangent and subnormal are equal then the length of tangent is equal to

• A. $y_1$
• B. $\sqrt{2}{y}_1$
• C. $\sqrt{2y_1}$
• D. None of these

Answer 1

The correct option is B $\sqrt{2}{y}_1$

Length of subtangent $=$ Length of subnormal
$y_1\times {\left(\frac{dx}{dy}\right)}_{(x_1,y_1)}=y_1\times {\left(\frac{dy}{dx}\right)}_{(x_1,y_1)}$
$\Rightarrow {\left(\frac{dy}{dx}\right)}^2=1$
$\therefore$ The length of tangent $={\frac{y_1{\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}}_{(x_1,y_1)}}{\left(\frac{dy}{dx}\right)}}_{(x_1,y_1)}$
$=y_1\times \sqrt{2}=\sqrt{2}\times y_1$
Hence, option 'B' is correct.