Question

If the length of the tangent drawn at the point (1,3) on the curve y = $3x^3$ is a, then the value of $9a^2$ is

• A. 82
• B. 81
• C. 729

Answer 1

The correct option is A 82

The length of the tangent at a point $(x_0,y_0)$ is given by:
$\left|y_0\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}\right|$

The given curve is $y=3x^3$

$\frac{dy}{dx}=9x^2$

At (1,3), $\frac{dy}{dx}=9\times 1^2=9$

$\therefore \frac{dx}{dy}=\frac{1}{9}$

Thus, the length of the tangent is $\left|y_0\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}\right|$

$=\left|3\sqrt{1+{\left(\frac{1}{9}\right)}^2}\right|$

$=\left|3\sqrt{(\frac{82}{81}}\right|$

$=\sqrt{\left(\frac{82\times 9}{81}\right)}$

$=\sqrt{\frac{82}{9}}$

Thus, $a=\sqrt{\frac{82}{9}}$

$a^2=\frac{82}{9}$

$9a^2=82$