Question

If the tangent at point $A(2,4)$ on the curve $y=x^3-2x^2+4$ cuts the $x$ -axis at point $T$, then the length of $AT$ is

• A. $\sqrt{17}$ units
• B. $2\sqrt{17}$ units
• C. $4\sqrt{17}$ units
• D. $\frac{1}{2}\sqrt{17}$ units

Answer 1

The correct option is A $\sqrt{17}$ units

$y=x^3-2x^2+4\Rightarrow \frac{dy}{dx}=3x^2-4x$

Slope of $T$ at $A(2,4)$, $m=3(2{)}^2-4(2)=4$

$AT$ = length of Tangent = $\left|y_1\sqrt{1+\frac{1}{m^2}}\right|$

$=\left|4\sqrt{1+\frac{1}{16}}\right|=\sqrt{17}$ units.