Question

A curve has equation $y=x(x-a)(x+a)$,where $a$ is a constant. Find the equations of the tangents to the graph at the points where it crosses the x-axis.

Answer 1

$y=x(x-a)(x+a)=x(x^2-a^2)=x^3-a^2{x}^2$

At x-axus, $y=0$

$\therefore$ $x=0,x=0,x=-a$

$(0,0)$, $(a,0)$, $(-a,0)$

$\frac{dy}{dx}=3x^2-2a{x}^2$

$y-0=0(x-0)$; $y-0=a^2(x-a)$

$y=\mathrm{0....}\left(i\right)$ ; $y=a^{2}x-a^3....\left(ii\right)$

$y-0=5a^2(x+a)$

$\Rightarrow$ $y=5a^{2}x+5a^3....\left(iii\right)$