The condition for the line y=mx+c to be a nor | vedclass

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(A) Given the equation of the parabola $y^{2}=4ax$. Let the parametric point on the parabola be $(at^{2}, 2at)$.The slope of the tangent at this point

Vedclass · Accepted Answer

$c=-2am-am^{3}$

Vedclass · Answer

(A) Given the equation of the parabola $y^{2}=4ax$. Let the parametric point on the parabola be $(at^{2}, 2at)$.The slope of the tangent at this point is $\frac{dy}{dx} = \frac{2a}{y} = \frac{2a}{2at} = \frac{1}{t}$.The slope of the normal is $-t$. Let the slope of the normal be $m$,so $m = -t$,which implies $t = -m$.The equation of the normal at point $(at^{2}, 2at)$ is $y - 2at = -t(x - at^{2})$.Substituting $t = -m$,we get $y - 2a(-m) = -(-m)(x - a(-m)^{2})$.$y + 2am = m(x - am^{2})$.$y + 2am = mx - am^{3}$.$y = mx - 2am - am^{3}$.Comparing this with the given line $y = mx + c$,we get $c = -2am - am^{3}$.

The condition for the line $y=mx+c$ to be a normal to the parabola $y^{2}=4ax$ is

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