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’(x) = $9x^2$

$\Rightarrow$ f’(1) = 9

$\Rightarrow$ m = 9

Now we will write the equation of tangent using slope(9) and one point(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. We get $9a^2=82$