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# how to remember equation of tangent . can you show how did you find out Equation of tangent at $(a{t_1}^2,2a{t_1})$ is

Equation of tangent at (at21,2at1) is t1y=x+at21-------(1) How did you get this

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A)
Equation of a parabola is

$y^2 = 4ax$

on differentiating w.r.t x  we get,

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

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

$\frac{dy}{dx}$ at $(a{t_1}^2,2a{t_1}) = \frac{2a}{2a{t_1}} = \frac{1}{{t_1}}$

Hence $m = \frac{1}{{t_1}}$

Equation of a tangent is $y-{y_1}=m(x-{x_1})$

$y-2a{t_1} = \frac{1}{t1}(x-a{t_1}^2)$

on simplifying we get

$y{t_1} -2a{t_1}^2 = x-a{t_1}^2$

that is  $y{t_1} = x + a{t_1}^2$

Hence proved.

Easy way to remember the equation of any tangent to a curve is $y-{y_1}=m(x-{x_1})$