Question

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

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Answer 1

Length of the tangent at a point is defined as the length of tangent between the given point and the point where it cuts x -axis. So, we will first find the tangent and find the point where it cuts the x-axis to find the length of the tangent.

We can find the slope of the tangent by taking derivative of y in the given equation

$f^{\prime }\left(x\right)=9x^2$

$\Rightarrow f^{\prime }\left(1\right)=9$

$\Rightarrow m=9$

Now we will write the equation of tangent using slope as 9 and point as (1,3)

$\Rightarrow y-3=9(x-1)$

To find the point where it cuts the x-axis, substitute $y=0$ in this

$\Rightarrow -3=9(x-1)$

$\Rightarrow x=\frac{2}{3}$

So the length of the tangent is the distance between the points $(\frac{2}{3},0)$ and $(1,3)$

$=\sqrt{{\left(\frac{1}{3}\right)}^2+3^2}=\sqrt{\left(\frac{1}{9}\right)+9}=\sqrt{\frac{82}{9}}$

This is given as ‘a’ in the question.

$\Rightarrow$ $a^2=82/9$

Hence, $9a^2=82$