Question

The length of tangent, subtangent, normal and subnormal for the curve $y=x^2+x-1$ at $\left(1,1\right)$ are $A,B,C$ and $D$ respectively, then their increasing order is

• A. $B,D,A,C$
• B. $B,A,C,D$
• C. $A,B,C,D$
• D. $B,A,D,C$

Answer 1

The correct option is D

$B,A,D,C$

Explanation for correct option:

Given equation of the curve is $y=x^2+x-1$

$$\begin{gathered} \therefore \frac{dy}{dx}=2x+1 \\ \Rightarrow {\left[\frac{dy}{dx}\right]}_{\left(1,1\right)}=3 \end{gathered}$$

Now $A$$=$Length of tangent at $\left(1,1\right)$

$A=\frac{\sqrt{1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2}}{{\displaystyle \frac{dy}{dx}}}=\frac{\sqrt{1+3^2}}{3}=\frac{\sqrt{10}}{3}$

Now $B$$=$Length of subtangent at $\left(1,1\right)$

$B=\left|y·\frac{dx}{dy}\right|=\frac{1}{3}$

Now $C$$=$Length of normal at $\left(1,1\right)$

$C=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}=\sqrt{1+3^2}=\sqrt{10}$

Now $D$$=$Length of subnormal at $\left(1,1\right)$

$D=\left|y·\frac{dy}{dx}\right|=3$

Thus we have $B<A<D<C$

Hence, option D is correct.