Question

The lengths of tangent, subtangent, normal and subnormal for the curve $y=x^{2}+x-1$ at $(1,1)$ 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

Answer: (D)

Given circle is $y=x^{2}+x-1$ and point $\left(x_{1}, y_{1}\right)=(1,1)$
$\therefore \frac{d y}{d x}=2 x+1$
At $(1,1),\,\,\, \frac{d y}{d x}=3=m$
Now, length of tangent
$A=\left|\frac{y_{1} \sqrt{1+m^{2}}}{m}\right|=\left|\frac{1 \sqrt{1+9}}{3}\right|=\frac{\sqrt{10}}{3}$
length of subtangent
$B=\left|\frac{y_{1}}{m}\right|=\frac{1}{3}$
length of normal
$C-\left|y_{1} \sqrt{1+m^{2}}\right|-|1 \sqrt{1+9}|-\sqrt{10}$
and length of subnormal $D=\left |y_{1} m\right|=3$
Now, increasing order is $B, A, D, C$.