Q106 / 160 · AP EAMCET 2007 · Mathematics

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
    $A, B, C, D$
  • Correct answer
    D
    $B, A, D, C$

Solution

Given circle is $y=x^2+x-1$ and point $\begin{aligned} & \left(x_1, y_1\right)=(1,1) \\ & \therefore \frac{d y}{d x}=2 x+1\end{aligned}$ At $(1,1), \quad \frac{d y}{d x}=3=m$ 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 $B, A, D, C$.

Hence, the correct answer is option: 'D'.

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