Question

find the equation of the evolute of the parabola y2=4ax​

Answer 1

Answer:

it's very easy

Step-by-step explanation:

Answer 2

Given:

y² = 4ax

To Find:

evaluate the parabola

Solution:

It is given that y² = 4ax

Now, putting x = at² , y = 2at

⇒ $\frac{dx}{dt}$ = 2at

⇒ $\frac{dy}{dt}$ = 2a

⇒ $\frac{d^{2} y}{dx^{2} }$ = $\frac{-1}{t^{2} } .\frac{dt}{dx}$

         = $\frac{-1}{2at^{3} }$

Coordinators of the center is equivalent to(x, y)

It is given as,

x = at² -[(1 + 1/t²)/(-1/2at³)] . 1/t                [x = x-(1+y,2)/y².y1]

  = 3at² + 2a  ..(i)

y = 2at+ (1+1/t²)/-1/2at³

  = -2at³ ..(ii)

Now, eliminating 't' from both (i) and (ii) equation,

⇒ x = 3at² + 2a

⇒ 3at² = x-2a

⇒ t = (x-2a/3a)$^{1/2}$

Then putting 't' in equation (ii)

⇒ y = -2a(x-2a/3a)$^{3/2}$

⇒ 27a³y² = 4a²(x - 2a)³

27ay² = 4(x-2a)³

Hence, this is the equation of parabola.