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

The correct option is D $B,A,D,C$

The equation of the curve is
$y=x^2+x-1$
$\therefore \frac{dy}{dx}=2x+1$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{(1,1)}=3$
Now, $A=$ Length of the tangent at $(1,1)$
$=\frac{\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}}{\frac{dy}{dx}}=\frac{\sqrt{1+3^2}}{3}=\frac{\sqrt{10}}{3}$
$B=$ Length of the subtangent at $(1,1)$
$=\frac{y}{\left(\frac{dy}{dx}\right)}=\frac{1}{3}$
$C=$ Length of the normal at $(1,1)$
$=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}=\sqrt{1+3^2}=\sqrt{10}$
$D=$ Length of the subnormal at $(1,1)$
$\Rightarrow D=y\frac{dy}{dx}=1\times 3=3$
Thus, we have $B<A<D<C$