TANGENTS AND NORMALS

# TANGENTS AND NORMALS

C
10 million points

Find the equation of the normal to the curve $y=(x^2+x+1)(x-3)$ at the point where it cuts the $x-axis$

TANGENTS AND
chegendungu

C
10 million points

$y=(x^2+x+1)(x-3)$

When $y=0$

$(x^2+x+1)(x-3)=0$

But $x^2+x+1=0$ has no real roots,

$\therefore x=+3$

$\therefore$ the curve cuts the $x-axis$ at $(3,0)$

$y=x^3-2x^2-2x-3$

$\therefore$ grad $t=3x^2-4x-2$

When $x=3$

grad $y=27-12-2=13$

The gradient of the tangent at $(3,0)$ is $+13$, therefore the gradient of the normal at

$(3,0)$ is $-\frac{1}{13}$ and its equation is

$\frac{y-0}{x-3}=-\frac{1}{13}$

$\therefore 13y=-x+3$

$\therefore$ the equation of the normal is $x+13y-3=0$

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