Question

Find the equation of the tangent at 't' on the curve x=a sin^3t,y=cos^3t

Answer 1

$\large\underline{\sf{Solution-}}$

Given parametric functions are

$\rm \: x = a {sin}^{3}t$

and

$\rm \: y = {cos}^{3}t$

So, Point of contact of tangent say P, is

$\rm \: Coordinates \: of \: P \: = \: ( {asin}^{3}t, \: {cos}^{3}t)$

Now, Consider

$\rm \: x = a {sin}^{3}t$

On differentiating both sides w. r. t. t, we get

$\rm \:\dfrac{d}{dt} x = a\dfrac{d}{dt} {sin}^{3}t$

$\rm \:\dfrac{dx}{dt} = 3a {sin}^{2}t \dfrac{d}{dt} sint$

$\rm \:\dfrac{dx}{dt} = 3a {sin}^{2}t cost$

Now, Consider

$\rm \: y = {cos}^{3}t$

On differentiating both sides w. r. t. t, we get

$\rm \: \dfrac{d}{dt}y = \dfrac{d}{dt} {cos}^{3}t$

$\rm \: \dfrac{dy}{dt} = 3 {cos}^{2}t \dfrac{d}{dt} cost$

$\rm \: \dfrac{dy}{dt} = - 3 {cos}^{2}tsint$

So,

$\rm \: \dfrac{dy}{dx} = \dfrac{dy}{dt} \div \dfrac{dx}{dt}$

$\rm \: = \: \dfrac{ - 3 {cos}^{2} t \: sint}{3a {sin}^{2} t \: cost}$

$\rm \: = \: - \dfrac{cost}{a \: sint}$

So,

$\rm \: Slope\:of\:tangent, \: m = \bigg(\dfrac{dy}{dx}\bigg)_{P} \: = \: - \dfrac{cost}{a \: sint}$

We know,

Equation of line which passes through the point (a, b) and having slope m is y - b = m(x - a)

So, equation of tangent at P is given by

$\rm \: y - {cos}^{3}t = - \dfrac{cost}{a \: sint}(x - {asin}^{3}t)$

$\rm \: (asint) y - asint {cos}^{3}t = - (cost)x + acost {sin}^{3}t$

$\rm \: (asint) y + (cost)x = acost {sin}^{3}t + asint {cos}^{3}t$

$\rm \: (asint) y + (cost)x = asint \: cost( {sin}^{2}t + {cos}^{2}t)$

$\rm \: (asint) y + (cost)x = asint \: cost( 1)$

$\rm \: (asint) y + (cost)x = asint \: cost$

Hence, the required equation of tangent is

$\rm\implies \:xcost \: + \: a \: sint \: y \: = \: a \: sint \: cost$

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Formula Used

$\rm \: \dfrac{d}{dx} {x}^{n} \: = \: {nx}^{n - 1}$

$\rm \: \dfrac{d}{dx} sinx \: = \: cosx$

$\rm \: \dfrac{d}{dx} cosx \: = \: - \: sinx$

ADDITIONAL INFORMATION

1. Let y = f(x) be any curve, then line which touches the curve y = f(x) exactly at one point say P is called tangent and that very point P, if we draw a perpendicular on tangent, that line is called normal to the curve at P.

2. If tangent is parallel to x - axis, its slope is 0.

3. If tangent is parallel to y - axis, its slope is not defined.

4. Two lines having slope M and m are parallel, iff M = m.

5. If two lines having slope M and m are perpendicular, iff Mm = - 1.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}$