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Let P (h,K) be any point on curve y=f...

Let P (h,K) be any point on curve `y=f(x).` Let tangent drawn at point P meets x-axis at T & normal at point P meets x-axis at N (as shown in figure) and m `=(dy)/(dx)]_()(h,k))` = shope of tangent.

(i) Length of tangent =PT `=|K| sqrt(1+(1)/(m^(2)))`
(ii) Length of Normal =PN +`|K| sqrt(1+m^(2))`
(iii) Length subtangent = Projection of segment PT on x-axis `=TM =|(k)/(m)|`
(iv) Length of subnormal =Projection of line segment PN on x-axis =MN `=|Km|`
Find the product of length of tangent and length of normal for the curve `y=x^(3)+3x^(2)+4x-1` at point x=0

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