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# Give answer! Equation of curves such that tangent and normal drawn at any point of which are of equal length, is

Equation of curves such that tangent and normal drawn at any point of which are of equal length, is

• Option 1)

$y =3x +c$

• Option 2)

$y =2x +c$

• Option 3)

$y =4x +c$

• Option 4)

$y =x +c$

Answers (1)
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As we have learnt,

Length of Normal -

$y\sqrt{1+\left ( \frac{dy}{dx} \right )^{2}}$

- wherein

$y\sqrt{1+\left ( \frac{dx}{dy} \right )^{2}}=y\sqrt{1+\left ( \frac{dy}{dx} \right )^{2}}$

Squaring both sides we have

$\left(\frac{dx}{dy} \right )^{2}=\left ( \frac{dy}{dx} \right )^{2} \\*\Rightarrow \frac{dx}{dy} = \pm \frac{dy}{dx} \\*\Rightarrow \frac{dy}{dx} = 1 \; or\;-1$

$\therefore$ Curves are either  $y = x + c \;or\; y = -x + c$

Option 1)

$y =3x +c$

Option 2)

$y =2x +c$

Option 3)

$y =4x +c$

Option 4)

$y =x +c$

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