Question

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

• A. Ordinate
• B. $\sqrt{2}|\text{ordinate}|$
• C. $\sqrt{\text{2 ordinate}}$
• D. $2\sqrt{\text{ordinate}}$

Answer 1

The correct option is B $\sqrt{2}|\text{ordinate}|$

Let $P(x_1,y_1)$ be a point on the curve $y=f\left(x\right)$. Then

The length of the tangent to the curve at point $P$ is

$\left|\frac{y_1}{m}\sqrt{1+m^2}\right|$

So, if the subtangent and subnormal are equal, then

$\frac{y_1}{m}=m{y}_1$

$m^2=1$

$m=\pm 1$

Thus, the length of the tangent will be

$=\left|\frac{y_1}{1}\sqrt{1+1}\right|$

$=\left|y_1\sqrt{2}\right|$

$=\sqrt{2}\left|y_1\right|$

Hence, this is the answer.