Question

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 ___

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Solution

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

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