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(9) and one point(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 23,0and (1,3) =√(13)2+32=√(19)+9=√829 This is given as ‘a’ in the question. We get 9a2=82

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