If the length of the tangent drawn at the point (1,3) on the curve y=3x3 is a, then find the value of 9a2
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)=9x2
⇒f′(1)=9
⇒m=9
Now we will write the equation of tangent using slope as 9 and point as (1,3)
⇒y−3=9(x−1)
To find the point where it cuts the x-axis, substitute y=0 in this
⇒−3=9(x−1)
⇒x=23
So the length of the tangent is the distance between the points (23,0) and (1,3)
=√(13)2+32=√(19)+9=√829
This is given as ‘a’ in the question.
⇒ a2=82/9
Hence, 9a2=82