If the lengths of the tangent,subtangent,norm | vedclass

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(D) Given the curve $y = x^2 + x - 1$ and the point $(1, 1)$.First,find the derivative $\frac{dy}{dx} = 2x + 1$.At the point $(1, 1)$,the slope $m = \

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$b, a, d, c$

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(D) Given the curve $y = x^2 + x - 1$ and the point $(1, 1)$.First,find the derivative $\frac{dy}{dx} = 2x + 1$.At the point $(1, 1)$,the slope $m = \frac{dy}{dx} = 2(1) + 1 = 3$.For a curve at a point $(x, y)$ with slope $m$:Length of tangent $a = |y| \sqrt{1 + \frac{1}{m^2}} = |1| \sqrt{1 + \frac{1}{9}} = \sqrt{\frac{10}{9}} = \frac{\sqrt{10}}{3} \approx 1.054$.Length of subtangent $b = |\frac{y}{m}| = |\frac{1}{3}| = 0.333$.Length of normal $c = |y| \sqrt{1 + m^2} = |1| \sqrt{1 + 9} = \sqrt{10} \approx 3.162$.Length of subnormal $d = |ym| = |1 	imes 3| = 3$.Comparing the values: $b = 0.333$,$a = 1.054$,$d = 3$,$c = 3.162$.Thus,the increasing order is $b < a < d < c$.

If the lengths of the tangent,subtangent,normal and subnormal for the curve $y=x^2+x-1$ at the point $(1,1)$ are $a, b, c$ and $d$ respectively,then their increasing order is

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