Question

If x=acost and y=bsint then find d2y/dx2

Answer 1

x=a cost
dx/dt=-asint
y=bsint
dy/dt=bcost
dy/dx=dy/dt/dx/dt=bcost/-acost=-b/acott
d2y/dx2=b/a×d/dt(cott)
d2y/dx2=-b/acosec^3t

Answer 2

Answer:

$\frac{ {d}^{2}y }{d {x}^{2} } = \frac{ - b \: {cosec}^{3}t }{ {a}^{2} }$

Step-by-step explanation:

If

$x = a \: cos \: t \\ \\ y = b \: sin \: t$

to find the value of

$\frac{ {d}^{2} y}{d {x}^{2} }$

firstly we have to find dy/dx

For that find dx/dt and dy/dt

$x = a \: cos \: t \\ \\ \frac{dx}{dt} = - a \: sin \: t \\ \\ y = b \: sin \: t \\ \\ \frac{dy}{dt} = b \: cos\: t \\ \\ \frac{dy}{dx} = \frac{dy}{dt} \times \frac{dt}{dx} = \frac{ b \: cos \: t}{ - a \: sin \: t} \\ \\ \frac{dy}{dt} = \frac{ - b}{a} cot \: t \\ \\ \frac{ {d}^{2}y }{d {x}^{2} } = \frac{ - b}{a} ( - {cosec}^{2} t). \frac{dt}{dx} \\ \\ = \frac{ b}{a} ( {cosec}^{2} t).( \frac{ - 1}{a \: sin \: t} ) \\ \\\frac{ {d}^{2}y }{d {x}^{2} } = \frac{ - b \: {cosec}^{3}t }{ {a}^{2} }$

Hope it helps you.