Question

Plz solve fast I need help

Answer 1

Step-by-step explanation:

We have,

$x = a {t}^{2} \: \: and \: \: y = 2at$

Now,

$t = \frac{y}{2a}$

Put this value in $x=at^2$

we get,

$x = a \bigg( \frac{y}{2a} \bigg)^{2}$

$\implies \: x = a \frac{ {y}^{2} }{4a ^{2} }$

$\implies \: x = \frac{ {y}^{2} }{4a}$

$\implies \: {y}^{2} =4ax$

Differentiating both sides w.r.t x, we get,

$\implies \: 2y. \frac{dy}{dx} =4a$

$\implies \: y. \frac{dy}{dx} =2a \: \: \: \: \:$

$\implies \: \frac{dy}{dx} = \frac{2a}{y} \: \: \: \: \:$

$\implies \: \frac{dx}{dy} = \frac{y}{2a} \: \: \: \: \:$

$\implies \: \frac{dx}{dy} = \frac{2at}{2a}$

$\implies \: \frac{dx}{dy} = t$

Differentiating w.r.t. y, we get,

$\implies \: \frac{d^{2} x}{dy ^{2} } = \frac{dt}{dy}$

$\implies \: \frac{d^{2} x}{dy ^{2} } = \frac{1}{2a}$