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
Answer & explanation
Correct answer: option 4
The equation of the curve is $y=x^2+x-1$
∴ $\frac{d y}{d x}=2 x+1 \Rightarrow\left(\frac{d y}{d x}\right)_{(1,1)}=3$
Now,
A = Length of the tangent at (1, 1)
$\Rightarrow A=\frac{\sqrt{1+\left(\frac{d y}{d x}\right)^2}}{\frac{d y}{d x}}=\frac{\sqrt{1+3^2}}{3}=\frac{\sqrt{10}}{3}$
B = Length of the subtangent at (1, 1)
$\Rightarrow B=\frac{y}{\frac{d x}{d y}}=\frac{1}{3}$
C = Length of the normal at (1, 1)
$\Rightarrow C =\sqrt{1+\left(\frac{d y}{d x}\right)^2}=\sqrt{1+3^2}=\sqrt{10}$
D = Length of the subnormal at (1, 1)
$\Rightarrow D =y \frac{d y}{d x}=1 \times 3=3$
Thus, we have B < A < D < C.