Find the slope of the tangent to the curve $x = a t^{2}, y = 2 a t$ at $t = 1$

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Solution

Consider the following equation.
$x = a t^{2}, y = 2 a t a t t = 1$
Differentiate both side w.r.t x and y we get.
$\frac{d x}{d t} = 2 a t . . . . . (1)$
$\frac{d y}{d t} = 2 a . . . . . . . . . . (2)$
Divide eq.(1)in eq. (2) we get.
$\frac{d y}{d x} = \frac{2 a t}{2 a} . . . . . . (3)$
$\frac{d y}{d x} = t$
Put the value of $^{'} t^{'}$ in eq. (3) we get.
$\frac{d y}{d x} = 1$
$d y = d x$

Hence, this is the correct answer

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